TOPICS
IN
ABSOLUTE
ANABELIAN
GEOMETRY
III:
GLOBAL
RECONSTRUCTION
ALGORITHMS
Shinichi
Mochizuki
November
2015
Abstract.
In
the
present
paper,
which
forms
the
third
part
of
a
three-part
series
on
an
algorithmic
approach
to
absolute
anabelian
geometry,
we
apply
the
ab-
solute
anabelian
technique
of
Belyi
cuspidalization
developed
in
the
second
part,
together
with
certain
ideas
contained
in
an
earlier
paper
of
the
author
concerning
the
category-theoretic
representation
of
holomorphic
structures
via
either
the
topologi-
cal
group
SL
2
(R)
or
the
use
of
“parallelograms,
rectangles,
and
squares”,
to
develop
a
certain
global
formalism
for
certain
hyperbolic
orbicurves
related
to
a
once-
punctured
elliptic
curve
over
a
number
field.
This
formalism
allows
one
to
construct
certain
canonical
rigid
integral
structures,
which
we
refer
to
as
log-shells,
that
are
obtained
by
applying
the
logarithm
at
various
primes
of
a
number
field.
More-
over,
although
each
of
these
local
logarithms
is
“far
from
being
an
isomorphism”
both
in
the
sense
that
it
fails
to
respect
the
ring
structures
involved
and
in
the
sense
[cf.
Frobenius
morphisms
in
positive
characteristic!]
that
it
has
the
effect
of
exhibiting
the
“mass”
represented
by
its
domain
as
a
“somewhat
smaller
collection
of
mass”
than
the
“mass”
represented
by
its
codomain,
this
global
formalism
allows
one
to
treat
the
logarithm
operation
as
a
global
operation
on
a
number
field
which
satisfies
the
property
of
being
an
“isomomorphism
up
to
an
appropriate
renormal-
ization
operation”,
in
a
fashion
that
is
reminiscent
of
the
isomorphism
induced
on
differentials
by
a
Frobenius
lifting,
once
one
divides
by
p.
More
generally,
if
one
thinks
of
number
fields
as
corresponding
to
positive
characteristic
hyperbolic
curves
and
of
once-punctured
elliptic
curves
on
a
number
field
as
corresponding
to
nilpotent
ordinary
indigenous
bundles
on
a
positive
characteristic
hyperbolic
curve,
then
many
aspects
of
the
theory
developed
in
the
present
paper
are
reminiscent
of
[the
positive
characteristic
portion
of]
p-adic
Teichmüller
theory.
Contents:
Introduction
§0.
Notations
and
Conventions
§1.
Galois-theoretic
Reconstruction
Algorithms
§2.
Archimedean
Reconstruction
Algorithms
§3.
Nonarchimedean
Log-Frobenius
Compatibility
§4.
Archimedean
Log-Frobenius
Compatibility
§5.
Global
Log-Frobenius
Compatibility
Appendix:
Complements
on
Complex
Multiplication
2000
Mathematical
Subject
Classification.
Primary
14H30;
Secondary
14H25.
Typeset
by
AMS-TEX
1
2
SHINICHI
MOCHIZUKI
Introduction
§I1.
Summary
of
Main
Results
§I2.
Fundamental
Naive
Questions
Concerning
Anabelian
Geometry
§I3.
Dismantling
the
Two
Combinatorial
Dimensions
of
a
Ring
§I4.
Mono-anabelian
Log-Frobenius
Compatibility
§I5.
Analogy
with
p-adic
Teichmüller
Theory
Acknowledgements
§I1.
Summary
of
Main
Results
Let
k
be
a
finite
extension
of
the
field
Q
p
of
p-adic
numbers
[for
p
a
prime
def
number];
k
an
algebraic
closure
of
k;
G
k
=
Gal(k/k).
Then
the
starting
point
of
the
theory
of
the
present
paper
lies
in
the
elementary
observation
that
although
the
p-adic
logarithm
×
log
k
:
k
→
k
[normalized
so
that
p
→
0]
is
not
a
ring
homomorphism,
it
does
satisfy
the
important
property
of
being
Galois-equivariant
[i.e.,
G
k
-equivariant].
def
In
a
similar
vein,
if
F
is
an
algebraic
closure
of
a
number
field
F
,
G
F
=
Gal(F
/F
),
and
k,
k
arise,
respectively,
as
the
completions
of
F
,
F
at
a
nonar-
chimedean
prime
of
F
,
then
although
the
map
log
k
does
not
extend,
in
any
natural
×
way,
to
a
map
F
→
F
[cf.
Remark
5.4.1],
it
does
extend
to
the
“disjoint
union
of
the
log
k
’s
at
all
the
nonarchimedean
primes
of
F
”
in
a
fashion
that
is
Galois-
equivariant
[i.e.,
G
F
-equivariant]
with
respect
to
the
natural
action
of
G
F
on
the
×
resulting
disjoint
unions
of
the
various
k
⊆
k.
Contemplation
of
the
elementary
observations
made
above
led
the
author
to
the
following
point
of
view:
The
fundamental
geometric
framework
in
which
the
logarithm
operation
should
be
understood
is
not
the
ring-theoretic
framework
of
scheme
theory,
but
rather
a
geometric
framework
based
solely
on
the
abstract
profinite
groups
G
k
,
G
F
[i.e.,
the
Galois
groups
involved],
i.e.,
a
framework
which
satisfies
the
key
property
of
being
“immune”
to
the
operation
of
applying
the
logarithm.
Such
a
group-theoretic
geometric
framework
is
precisely
what
is
furnished
by
the
enhancement
of
absolute
anabelian
geometry
—
which
we
shall
refer
to
as
mono-
anabelian
geometry
—
that
is
developed
in
the
present
paper.
This
enhancement
may
be
thought
of
as
a
natural
outgrowth
of
the
algorithm-
based
approach
to
absolute
anabelian
geometry,
which
forms
the
unifying
theme
[cf.
the
Introductions
to
[Mzk20],
[Mzk21]]
of
the
three-part
series
of
which
the
present
paper
constitutes
the
third,
and
final,
part.
From
the
point
of
view
of
the
present
paper,
certain
portions
of
the
theory
and
results
developed
in
earlier
papers
of
the
present
series
—
most
notably,
the
theory
of
Belyi
cuspidalizations
developed
in
TOPICS
IN
ABSOLUTE
ANABELIAN
GEOMETRY
III
3
[Mzk21],
§3
—
are
relevant
to
the
theory
of
the
present
paper
partly
because
of
their
logical
necessity
in
the
proofs,
and
partly
because
of
their
philosophical
relevance
[cf.,
especially,
the
discussion
of
“hidden
endomorphisms”
in
the
Introduction
to
[Mzk21];
the
theory
of
[Mzk21],
§2].
Note
that
a
ring
may
be
thought
of
as
a
mathematical
object
that
consists
of
“two
combinatorial
dimensions”,
corresponding
to
its
additive
structure,
which
we
shall
denote
by
the
symbol
,
and
its
multiplicative
structure,
which
we
shall
denote
by
the
symbol
[cf.
Remark
5.6.1,
(i),
for
more
details].
One
way
to
understand
the
failure
of
the
logarithm
to
be
compatible
with
the
ring
structures
involved
is
as
a
manifestation
of
the
fact
that
the
logarithm
has
the
effect
of
“tinkering
with,
or
dismantling,
this
two-dimensional
structure”.
Such
a
dismantling
operation
cannot
be
understood
within
the
framework
of
ring
[or
scheme]
theory.
That
is
to
say,
it
may
only
be
understood
from
the
point
of
view
of
a
geometric
framework
that
“lies
essentially
outside”,
or
“is
neutral
with
respect
to”,
this
two-dimensional
structure
[cf.
the
illustration
of
Remark
5.10.2,
(iii)].
One
important
property
of
the
p-adic
logarithm
log
k
discussed
above
is
that
the
image
log
k
(O
k
×
)
⊆
k
—
which
is
compact
—
may
be
thought
of
as
defining
a
sort
of
canonical
rigid
integral
structure
on
k.
In
the
present
paper,
we
shall
refer
to
the
“canonical
rigid
integral
structures”
obtained
in
this
way
as
log-shells.
Note
that
the
image
log
k
(k
×
)
of
k
×
via
log
k
is,
like
log
k
(O
k
×
)
[but
unlike
k
×
!],
compact.
That
is
to
say,
the
operation
of
applying
the
p-adic
logarithm
may
be
thought
of
as
a
sort
of
“compression”
operation
that
exhibits
the
“mass”
represented
by
its
domain
as
a
“somewhat
smaller
collection
of
mass”
than
the
“mass”
represented
by
its
codomain.
In
this
sense,
the
p-adic
logarithm
is
reminiscent
of
the
Frobenius
morphism
in
positive
characteristic
[cf.
Remark
3.6.2
for
more
details].
In
particular,
this
“compressing
nature”
of
the
p-adic
logarithm
may
be
thought
of
as
being
one
that
lies
in
sharp
contrast
with
the
nature
of
an
étale
morphism.
This
point
of
view
is
reminiscent
of
the
discussion
of
the
“fundamental
dichotomy”
between
“Frobenius-
like”
and
“étale-like”
structures
in
the
Introduction
of
[Mzk16].
In
the
classical
p-adic
theory,
the
notion
of
a
Frobenius
lifting
[cf.
the
theory
of
[Mzk1],
[Mzk4]]
may
be
thought
of
as
forming
a
bridge
between
the
two
sides
of
this
dichotomy
[cf.
the
discussion
of
mono-theta
environments
in
the
Introduction
to
[Mzk18]!]
—
that
is
to
say,
a
Frobenius
lifting
is,
on
the
one
hand,
literally
a
lifting
of
the
Frobenius
morphism
in
positive
characteristic
and,
on
the
other
hand,
tends
to
satisfy
the
property
of
being
étale
in
characteristic
zero,
i.e.,
of
inducing
an
isomorphism
on
differentials,
once
one
divides
by
p.
In
a
word,
the
theory
developed
in
the
present
paper
may
be
summarized
as
follows:
The
thrust
of
the
theory
of
the
present
paper
lies
in
the
development
of
a
formalism,
via
the
use
of
ring/scheme
structures
reconstructed
via
mono-
anabelian
geometry,
in
which
the
“dismantling/compressing
nature”
of
the
logarithm
operation
discussed
above
[cf.
the
Frobenius
morphism
in
pos-
itive
characteristic]
is
“reorganized”
in
an
abstract
combinatorial
fashion
4
SHINICHI
MOCHIZUKI
that
exhibits
the
logarithm
as
a
global
operation
on
a
number
field
which,
moreover,
is
a
sort
of
“isomomorphism
up
to
an
appropriate
renormalization
operation”
[cf.
the
isomorphism
induced
on
differen-
tials
by
a
Frobenius
lifting,
once
one
divides
by
p].
One
important
aspect
of
this
theory
is
the
analogy
between
this
theory
and
[the
positive
characteristic
portion
of]
p-adic
Teichmüller
theory
[cf.
§I5
below],
in
which
the
“naive
pull-back”
of
an
indigenous
bundle
by
Frobenius
never
yields
a
bundle
isomorphic
to
the
original
indigenous
bundle,
but
the
“renormalized
Frobe-
nius
pull-back”
does,
in
certain
cases,
allow
one
to
obtain
an
output
bundle
that
is
isomorphic
to
the
original
input
bundle.
At
a
more
detailed
level,
the
main
results
of
the
present
paper
may
be
sum-
marized
as
follows:
In
§1,
we
develop
the
absolute
anabelian
algorithms
that
will
be
necessary
in
our
theory.
In
particular,
we
obtain
a
semi-absolute
group-theoretic
recon-
struction
algorithm
[cf.
Theorem
1.9,
Corollary
1.10]
for
hyperbolic
orbicurves
of
strictly
Belyi
type
[cf.
[Mzk21],
Definition
3.5]
over
sub-p-adic
fields
—
i.e.,
such
as
number
fields
and
nonarchimedean
completions
of
number
fields
—
that
is
func-
torial
with
respect
to
base-change
of
the
base
field.
Moreover,
we
observe
that
the
only
“non-elementary”
ingredient
of
these
algorithms
is
the
technique
of
Belyi
cuspidalization
developed
in
[Mzk21],
§3,
which
depends
on
the
main
results
of
[Mzk5]
[cf.
Remark
1.11.3].
If
one
eliminates
this
non-elementary
ingredient
from
these
algorithms,
then,
in
the
case
of
function
fields,
one
obtains
a
very
elemen-
tary
semi-absolute
group-theoretic
reconstruction
algorithm
[cf.
Theorem
1.11],
which
is
valid
over
somewhat
more
general
base
fields,
namely
base
fields
which
are
“Kummer-faithful”
[cf.
Definition
1.5].
The
results
of
§1
are
of
interest
as
anabelian
results
in
their
own
right,
independent
of
the
theory
of
later
portions
of
the
present
paper.
For
instance,
it
is
hoped
that
elementary
results
such
as
Theorem
1.11
may
be
of
use
in
introductions
to
anabelian
geometry
for
advanced
undergraduates
or
non-specialists
[cf.
[Mzk8],
§1].
In
§2,
we
develop
an
archimedean
—
i.e.,
complex
analytic
—
analogue
of
the
theory
of
§1.
One
important
theme
in
this
theory
is
the
definition
of
“archimedean
structures”
which,
like
profinite
Galois
groups,
are
“immune
to
the
ring
structure-
dismantling
and
compressing
nature
of
the
logarithm”.
For
instance,
the
notion
that
constitutes
the
archimedean
counterpart
to
the
notion
of
a
profinite
Galois
group
is
the
notion
of
an
Aut-holomorphic
structure
[cf.
Definition
2.1;
Propo-
sition
2.2;
Corollary
2.3],
which
was
motivated
by
the
category-theoretic
approach
to
holomorphic
structures
via
the
use
of
the
topological
group
SL
2
(R)
given
in
[Mzk14],
§1.
In
this
context,
one
central
fact
is
the
rather
elementary
observation
that
the
group
of
holomorphic
or
anti-holomorphic
automorphisms
of
the
unit
disc
in
the
complex
plane
is
commensurably
terminal
[cf.
[Mzk20],
§0]
in
the
group
of
self-homeomorphisms
of
the
unit
disc
[cf.
Proposition
2.2,
(ii)].
We
also
give
an
“al-
gorithmic
refinement”
of
the
“parallelograms,
rectangles,
squares
approach”
of
[Mzk14],
§2
[cf.
Propositions
2.5,
2.6].
By
combining
these
two
approaches
and
applying
the
technique
of
elliptic
cuspidalization
developed
in
[Mzk21],
§3,
we
ob-
tain
a
certain
reconstruction
algorithm
[cf.
Corollary
2.7]
for
the
“local
linear
holomorphic
structure”
of
an
Aut-holomorphic
orbispace
arising
from
an
el-
liptically
admissible
[cf.
[Mzk21],
Definition
3.1]
hyperbolic
orbicurve,
which
is
TOPICS
IN
ABSOLUTE
ANABELIAN
GEOMETRY
III
5
compatible
with
the
global
portion
of
the
Galois-theoretic
theory
of
§1
[cf.
Corollaries
2.8,
2.9].
In
§3,
§4,
we
develop
the
category-theoretic
formalism
—
centering
around
the
notions
of
observables,
telecores,
and
cores
[cf.
Definition
3.5]
—
that
are
applied
to
express
the
compatibility
of
the
“mono-anabelian”
construction
algo-
rithms
of
§1
[cf.
Corollary
3.6]
and
§2
[cf.
Corollary
4.5]
with
the
“log-Frobenius
functor
log”
[in
essence,
a
version
of
the
usual
“logarithm”
at
the
various
nonar-
chimedean
and
archimedean
primes
of
a
number
field].
We
also
study
the
failure
of
log-Frobenius
compatibility
that
occurs
if
one
attempts
to
take
the
“conventional
anabelian”
—
which
we
shall
refer
to
as
“bi-anabelian”
—
approach
to
the
situation
[cf.
Corollary
3.7].
Finally,
in
the
remarks
following
Corollaries
3.6,
3.7,
we
discuss
in
detail
the
meaning
of
the
various
new
category-theoretic
notions
that
are
intro-
duced,
as
well
as
the
various
aspects
of
the
analogy
between
these
notions,
in
the
context
of
Corollaries
3.6,
3.7,
and
the
classical
p-adic
theory
of
the
MF
∇
-objects
of
[Falt].
In
§5,
we
develop
a
global
formalism
over
number
fields
in
which
we
study
the
canonical
rigid
integral
structures
—
i.e.,
log-shells
—
that
are
obtained
by
applying
the
log-Frobenius
compatibility
discussed
in
§3,
§4.
These
log-shells
satisfy
the
following
important
properties:
(L1)
a
log-shell
is
compact
and
hence
of
finite
“log-volume”
[cf.
Corollary
5.10,
(i)];
(L2)
the
log-volumes
of
(L1)
are
compatible
with
application
of
the
log-
Frobenius
functor
[cf.
Corollary
5.10,
(ii)];
(L3)
log-shells
are
compatible
with
the
operation
of
“panalocalization”,
i.e.,
the
operation
of
restricting
to
the
disjoint
union
of
the
various
primes
of
a
number
field
in
such
a
way
that
one
“forgets”
the
global
structure
of
the
number
field
[cf.
Corollary
5.5;
Corollary
5.10,
(iii)];
(L4)
log-shells
are
compatible
with
the
operation
of
“mono-analyticization”,
i.e.,
the
operation
of
“disabling
the
rigidity”
of
one
of
the
“two
combina-
torial
dimensions”
of
a
ring,
an
operation
that
corresponds
to
allowing
“Teichmüller
dilations”
in
complex
and
p-adic
Teichmüller
theory
[cf.
Corollary
5.10,
(iv)].
In
particular,
we
note
that
property
(L3)
may
be
thought
of
as
a
rigidity
property
for
certain
global
arithmetic
line
bundles
[more
precisely,
the
trivial
arithmetic
line
bundle
—
cf.
Remarks
5.4.2,
5.4.3]
that
is
analogous
to
the
very
strong
—
i.e.,
by
comparison
to
the
behavior
of
arbitrary
vector
bundles
on
a
curve
—
rigidity
prop-
erties
satisfied
by
MF
∇
-objects
with
respect
to
Zariski
localization.
Such
rigidity
properties
may
be
thought
of
as
a
sort
of
“freezing
of
integral
structures”
with
respect
to
Zariski
localization
[cf.
Remark
5.10.2,
(i)].
Finally,
we
discuss
in
some
detail
[cf.
Remark
5.10.3]
the
analogy
—
centering
around
the
correspondence
number
field
F
once-punctured
ell.
curve
X
over
F
←→
hyperbolic
curve
C
in
pos.
char.
←→
nilp.
ord.
indig.
bundle
P
over
C
—
between
the
theory
of
the
present
paper
[involving
hyperbolic
orbicurves
related
to
once-punctured
elliptic
curves
over
a
number
field]
and
the
p-adic
Teichmüller
6
SHINICHI
MOCHIZUKI
theory
of
[Mzk1],
[Mzk4]
[involving
nilpotent
ordinary
indigenous
bundles
over
hyperbolic
curves
in
positive
characteristic].
Finally,
in
an
Appendix
to
the
present
paper,
we
expose
the
portion
of
the
well-known
theory
of
abelian
varieties
with
complex
multiplication
[cf.,
e.g.,
[Lang-
CM],
[Milne-CM],
for
more
details]
that
underlies
the
observation
“(∗
CM
)”
related
to
the
author
by
A.
Tamagawa
[cf.
[Mzk20],
Remark
3.8.1].
In
particular,
we
verify
that
this
observation
(∗
CM
)
does
indeed
hold.
This
implies
that
the
observation
“(∗
A-qLT
)”
discussed
in
[Mzk20],
Remark
3.8.1,
also
holds,
and
hence,
in
particular,
that
the
hypothesis
of
[Mzk20],
Corollary
3.9,
to
the
effect
that
“either
(∗
A-qLT
)
or
(∗
CM
)
holds”
may
be
eliminated
[i.e.,
that
[Mzk20],
Corollary
3.9,
holds
uncondi-
tionally].
Although
the
content
of
this
Appendix
is
not
directly
technically
related
to
the
remainder
of
the
present
paper,
the
global
arithmetic
nature
of
the
content
of
this
Appendix,
as
well
as
the
accompanying
discussion
of
the
relationship
of
this
global
content
with
considerations
in
p-adic
Hodge
theory,
is
closely
related
in
spirit
to
the
analogies
between
the
content
of
the
remainder
of
the
present
paper
and
the
theory
of
earlier
papers
in
the
present
series
of
papers,
i.e.,
more
precisely,
[Mzk20],
§3;
[Mzk21],
§2.
§I2.
Fundamental
Naive
Questions
Concerning
Anabelian
Geometry
One
interesting
aspect
of
the
theory
of
the
present
paper
is
that
it
is
intimately
related
to
various
fundamental
questions
concerning
anabelian
geometry
that
are
frequently
posed
by
newcomers
to
the
subject.
Typical
examples
of
these
fundamen-
tal
questions
are
the
following:
(Q1)
Why
is
it
useful
or
meaningful
to
study
anabelian
geometry
in
the
first
place?
(Q2)
What
exactly
is
meant
by
the
term
“group-theoretic
reconstruction”
in
discussions
of
anabelian
geometry?
(Q3)
What
is
the
significance
of
studying
anabelian
geometry
over
mixed-
characteristic
local
fields
[i.e.,
p-adic
local
fields]
as
opposed
to
num-
ber
fields?
(Q4)
Why
is
birational
anabelian
geometry
insufficient
—
i.e.,
what
is
the
significance
of
studying
the
anabelian
geometry
of
hyperbolic
curves,
as
opposed
to
their
function
fields?
In
fact,
the
answers
to
these
questions
that
are
furnished
by
the
theory
of
the
present
paper
are
closely
related.
As
was
discussed
in
§I1,
the
answer
to
(Q1),
from
the
point
of
view
of
the
present
paper,
is
that
anabelian
geometry
—
more
specifically,
“mono-anabelian
geometry”
—
provides
a
framework
that
is
sufficiently
well-endowed
as
to
contain
“data
reminiscent
of
the
data
constituted
by
various
scheme-theoretic
structures”,
but
has
the
virtue
of
being
based
not
on
ring
structures,
but
rather
on
profinite
[Galois]
groups,
which
are
“neutral”
with
respect
to
the
operation
of
taking
the
logarithm.
TOPICS
IN
ABSOLUTE
ANABELIAN
GEOMETRY
III
7
The
answer
to
(Q2)
is
related
to
the
algorithmic
approach
to
absolute
anabelian
geometry
taken
in
the
present
three-part
series
[cf.
the
Introduction
to
[Mzk20]].
That
is
to
say,
typically,
in
discussions
concerning
“Grothendieck
Conjecture-type
fully
faithfulness
results”
[cf.,
e.g.,
[Mzk5]]
the
term
“group-theoretic
reconstruc-
tion”
is
defined
simply
to
mean
“preserved
by
an
arbitrary
isomorphism
between
the
étale
fundamental
groups
of
the
two
schemes
under
consideration”.
This
point
of
view
will
be
referred
to
in
the
present
paper
as
“bi-anabelian”.
By
contrast,
the
algorithmic
approach
to
absolute
anabelian
geometry
involves
the
development
of
“software”
whose
input
data
consists
solely
of,
for
instance,
a
single
abstract
profinite
group
[that
just
happens
to
be
isomorphic
to
the
étale
fundamental
group
of
a
scheme],
and
whose
output
data
consists
of
various
structures
reminiscent
of
scheme
theory
[cf.
the
Introduction
to
[Mzk20]].
This
point
of
view
will
be
referred
to
in
the
present
paper
as
“mono-anabelian”.
Here,
the
mono-anabelian
“soft-
ware”
is
required
to
be
functorial,
e.g.,
with
respect
to
isomorphisms
of
profinite
groups.
Thus,
it
follows
formally
that
“mono-anabelian”
=⇒
“bi-anabelian”
[cf.
Remark
1.9.8].
On
the
other
hand,
although
it
is
difficult
to
formulate
such
issues
completely
precisely,
the
theory
of
the
present
paper
[cf.,
especially,
§3]
sug-
gests
strongly
that
the
opposite
implication
should
be
regarded
as
false.
That
is
to
say,
whereas
the
mono-anabelian
approach
yields
a
framework
that
is
“neutral”
with
respect
to
the
operation
of
taking
the
logarithm,
the
bi-anabelian
approach
fails
to
yield
such
a
framework
[cf.
Corollaries
3.6,
3.7,
and
the
following
remarks;
§I4
below].
Here,
we
pause
to
remark
that,
in
fact,
although,
historically
speaking,
many
theorems
were
originally
formulated
in
a
“bi-anabelian”
fashion,
careful
inspec-
tion
of
their
proofs
typically
leads
to
the
recovery
of
“mono-anabelian
algorithms”.
Nevertheless,
since
formulating
theorems
in
a
“mono-anabelian”
fashion,
as
we
have
attempted
to
do
in
the
present
paper
[and
more
generally
in
the
present
three-part
series,
but
cf.
the
final
portion
of
the
Introduction
to
[Mzk21]],
can
be
quite
cumber-
some
—
and
indeed
is
one
of
the
main
reasons
for
the
unfortunately
lengthy
nature
of
the
present
paper!
—
it
is
often
convenient
to
formulate
final
theorems
in
a
“bi-
anabelian”
fashion.
On
the
other
hand,
we
note
that
the
famous
Neukirch-Uchida
theorem
on
the
anabelian
nature
of
number
fields
appears
to
be
one
important
counterexample
to
the
above
remark.
That
is
to
say,
to
the
author’s
knowledge,
proofs
of
this
result
never
yield
“explicit
mono-anabelian
reconstruction
algorithms
of
the
given
number
field”;
by
contrast,
Theorem
1.9
of
the
present
paper
does
give
such
an
explicit
construction
of
the
“given
number
field”
[cf.
Remark
1.9.5].
Another
interesting
aspect
of
the
algorithmic
approach
to
anabelian
geometry
is
that
one
may
think
of
the
“software”
constituted
by
such
algorithms
as
a
sort
of
“combinatorialization”
of
the
original
schemes
[cf.
Remark
1.9.7].
This
point
of
view
is
reminiscent
of
the
operation
of
passing
from
a
“scheme-theoretic”
MF
∇
-
object
to
an
associated
Galois
representation,
as
well
as
the
general
theme
in
various
papers
of
the
author
concerning
a
“category-theoretic
approach
to
scheme
theory”
[cf.,
e.g.,
[Mzk13],
[Mzk14],
[Mzk16],
[Mzk17],
[Mzk18]]
of
“extracting
from
scheme-
theoretic
arithmetic
geometry
the
abstract
combinatorial
patterns
that
underlie
the
scheme
theory”.
8
SHINICHI
MOCHIZUKI
The
answer
to
(Q3)
provided
by
the
theory
of
the
present
paper
is
that
the
absolute
p-adic
[mono-]anabelian
results
of
§1
underlie
the
panalocalizability
of
log-shells
discussed
in
§I1
[cf.
property
(L3)].
Put
another
way,
these
results
imply
that
the
“geometric
framework
immune
to
the
application
of
the
logarithm”
—
i.e.,
immune
to
the
dismantling
of
the
“”
and
“”
dimensions
of
a
ring
—
discussed
in
§I1
may
be
applied
locally
at
each
prime
of
a
number
field
regarded
as
an
isolated
entity,
i.e.,
without
making
use
of
the
global
structure
of
the
number
field
—
cf.
the
discussion
of
“freezing
of
integral
structures”
with
respect
to
Zariski
localization
in
Remark
5.10.2,
(i).
For
more
on
the
significance
of
the
operation
of
passing
“
”
in
the
context
of
nonarchimedean
log-shells
—
i.e.,
the
operation
of
passing
“O
k
×
log
k
(O
k
×
)”
—
we
refer
to
the
discussion
of
nonarchimedean
log-shells
in
§I3
below.
The
answer
to
(Q4)
furnished
by
the
theory
of
the
present
paper
[cf.
Remark
1.11.4]
—
i.e.,
one
fundamental
difference
between
birational
anabelian
geometry
and
the
anabelian
geometry
of
hyperbolic
curves
—
is
that
[unlike
spectra
of
func-
tion
fields!]
“most”
hyperbolic
curves
admit
“cores”
[in
the
sense
of
[Mzk3],
§3;
[Mzk10],
§2],
which
may
be
thought
of
as
a
sort
of
abstract
“covering-theoretic”
analogue
[cf.
Remark
1.11.4,
(ii)]
of
the
notion
of
a
“canonical
rigid
integral
struc-
ture”
[cf.
the
discussion
of
log-shells
in
§I1].
Moreover,
if
one
attempts
to
work
with
the
Galois
group
of
a
function
field
supplemented
by
some
additional
structure
such
as
the
set
of
cusps
—
arising
from
scheme
theory!
—
that
determines
a
hyperbolic
curve
structure,
then
one
must
sacrifice
the
crucial
mono-anabelian
nature
of
one’s
reconstruction
algorithms
[cf.
Remarks
1.11.5;
3.7.7,
(ii)].
Finally,
we
observe
that
there
certainly
exist
many
“fundamental
naive
ques-
tions”
concerning
anabelian
geometry
for
which
the
theory
of
the
present
paper
does
not
furnish
any
answers.
Typical
examples
of
such
fundamental
questions
are
the
following:
(Q5)
What
is
the
significance
of
studying
the
anabelian
geometry
of
proper
hyperbolic
curves,
as
opposed
to
affine
hyperbolic
curves?
(Q6)
What
is
the
significance
of
studying
pro-Σ
[where
Σ
is
some
nonempty
set
of
prime
numbers]
anabelian
geometry,
as
opposed
to
profinite
anabelian
geometry
[cf.,
e.g.,
Remark
3.7.6
for
a
discussion
of
why
pro-Σ
anabelian
geometry
is
ill-suited
to
the
needs
of
the
theory
of
the
present
paper]?
(Q7)
What
is
the
significance
of
studying
anabelian
geometry
in
positive
characteristic,
e.g.,
over
finite
fields?
It
would
certainly
be
of
interest
if
further
developments
could
shed
light
on
these
questions.
§I3.
Dismantling
the
Two
Combinatorial
Dimensions
of
a
Ring
As
was
discussed
in
§I1,
a
ring
may
be
thought
of
as
a
mathematical
object
that
consists
of
“two
combinatorial
dimensions”,
corresponding
to
its
additive
structure
and
its
multiplicative
structure
[cf.
Remark
5.6.1,
(i)].
When
the
ring
un-
der
consideration
is
a
[say,
for
simplicity,
totally
imaginary]
number
field
F
or
TOPICS
IN
ABSOLUTE
ANABELIAN
GEOMETRY
III
9
a
mixed-characteristic
nonarchimedean
local
field
k,
these
two
combinatorial
dimensions
may
also
be
thought
of
as
corresponding
to
the
two
cohomological
dimensions
of
the
absolute
Galois
groups
G
F
,
G
k
of
F
,
k
[cf.
[NSW],
Proposition
8.3.17;
[NSW],
Theorem
7.1.8,
(i)].
In
a
similar
vein,
when
the
ring
under
con-
sideration
is
a
complex
archimedean
field
k
(
∼
=
C),
then
the
two
combinatorial
dimensions
of
k
may
also
be
thought
of
as
corresponding
to
the
two
topological
—
i.e.,
real
—
dimensions
of
the
underlying
topological
space
of
the
topological
group
k
×
.
Note
that
in
the
case
where
the
local
field
k
is
nonarchimedean
(respec-
tively,
archimedean),
precisely
one
of
the
two
cohomological
(respectively,
real)
dimensions
of
G
k
(respectively,
k
×
)
—
namely,
the
dimension
corresponding
to
the
·
Fr
[generated
by
the
Frobenius
element]
(re-
maximal
unramified
quotient
G
k
Z
spectively,
the
topological
subgroup
of
units
S
1
∼
=
O
k
×
⊆
k
×
)
is
rigid
with
respect
to,
say,
automorphisms
of
the
topological
group
G
k
(respectively,
k
×
),
while
the
other
dimension
—
namely,
the
dimension
corresponding
to
the
inertia
subgroup
I
k
⊆
G
k
(respectively,
the
value
group
k
×
R
>0
)
—
is
not
rigid
[cf.
Remark
1.9.4].
[In
the
nonarchimedean
case,
this
phenomenon
is
discussed
in
more
detail
in
[NSW],
the
Closing
Remark
preceding
Theorem
12.2.7.]
Thus,
each
of
the
various
nonarchimedean
“G
k
’s”
and
archimedean
“k
×
’s”
that
arise
at
the
various
primes
of
a
number
field
may
be
thought
of
as
being
a
sort
of
“arithmetic
G
m
”
—
i.e.,
an
abstract
arithmetic
“cylinder”
—
that
decomposes
into
a
[twisted]
product
of
·
Fr,
k
×
R
>0
]
“units”
[i.e.,
I
k
⊆
G
k
,
O
k
×
⊆
k
×
]
and
value
group
[i.e.,
G
k
Z
‘arithmetic
G
m
’
⎛
⎜
⎜
⎝
‘units’
‘×’
⎞
⎟
⎟
⎠
∼
→
‘×’
‘value
group’
⎛
⎞
⎜
⎜
⎝
⎟
⎟
⎠
with
the
property
that
one
of
these
two
factors
is
rigid,
while
the
other
is
not.
Here,
it
is
interesting
to
note
that
the
correspondence
between
units/value
group
and
rigid/non-rigid
differs
[i.e.,
“goes
in
the
opposite
direction”]
in
the
nonarchimedean
and
archimedean
cases.
This
phenomenon
is
reminiscent
of
the
product
formula
in
elementary
number
theory,
as
well
as
of
the
behavior
of
the
log-Frobenius
functor
log
at
nonarchimedean
versus
archimedean
primes
[cf.
Remark
4.5.2;
the
discussion
of
log-shells
in
the
final
portion
of
the
present
§I3].
∼
C
×
→
G
k
→
∼
rigid
S
1
×
non-rigid
I
k
non-rigid
R
>0
rigid
·
Fr
Z
On
the
other
hand,
the
perfection
of
the
topological
group
obtained
as
the
image
of
the
non-rigid
portion
I
k
in
the
abelianization
G
ab
k
of
G
k
is
naturally
iso-
morphic,
by
local
class
field
theory,
to
k.
Moreover,
by
the
theory
of
[Mzk2],
the
10
SHINICHI
MOCHIZUKI
decomposition
of
this
copy
of
k
[i.e.,
into
sets
of
elements
with
some
given
p-adic
val-
uation]
determined
by
the
p-adic
valuation
on
k
may
be
thought
of
as
corresponding
to
the
ramification
filtration
on
G
k
and
is
precisely
the
data
that
is
“deformed”
by
automorphisms
of
k
that
do
not
arise
from
field
automorphisms.
That
is
to
say,
this
aspect
of
the
non-rigidity
of
G
k
is
quite
reminiscent
of
the
non-rigidity
of
the
topological
group
R
>0
[i.e.,
of
the
non-rigidity
of
the
structure
on
this
topological
group
arising
from
the
usual
archimedean
valuation
on
R,
which
determines
an
isomorphism
between
this
topological
group
and
some
“fixed,
standard
copy”
of
R
>0
].
In
this
context,
one
of
the
first
important
points
of
the
“mono-anabelian
the-
ory”
of
§1,
§2
of
the
present
paper
is
that
if
one
supplements
a(n)
nonarchimedean
G
k
(respectively,
archimedean
k
×
)
with
the
data
arising
from
a
hyperbolic
or-
bicurve
[which
satisfies
certain
properties
—
cf.
Corollaries
1.10,
2.7],
then
this
supplementary
data
has
the
effect
of
rigidifying
both
dimensions
of
G
k
(re-
spectively,
k
×
).
In
the
case
of
[a
nonarchimedean]
G
k
,
this
data
consists
of
the
outer
action
of
G
k
on
the
profinite
geometric
fundamental
group
of
the
hyperbolic
orbicurve;
in
the
case
of
[an
archimedean]
k
×
,
this
data
consists,
in
essence,
of
the
various
local
actions
of
open
neighborhoods
of
the
origin
of
k
×
on
the
squares
or
rectangles
[that
lie
in
the
underlying
topological
[orbi]space
of
the
Riemann
[orbi]surface
determined
by
the
hyperbolic
[orbi]curve]
that
encode
the
holomor-
phic
structure
of
the
Riemann
[orbi]surface
[cf.
the
theory
of
[Mzk14],
§2].
Here,
it
is
interesting
to
note
that
these
“rigidifying
actions”
are
reminiscent
of
the
discussion
of
“hidden
endomorphisms”
in
the
Introduction
to
[Mzk21],
as
well
as
of
the
discussion
of
“intrinsic
Hodge
theory”
in
the
context
of
p-adic
Teichmüller
theory
in
[Mzk4],
§0.10.
⎛
⎜
⎜
⎝
⎞
⎟
⎟
⎠
C
×
G
k
rigidify
‘’
geometric
data
representing
hyp.
curve
rectangles/squares
on
hyp.
Riemann
surface
out
profinite
geometric
fund.
gp.
of
hyp.
curve
Thus,
in
summary,
the
“rigidifying
actions”
discussed
above
may
be
thought
of
as
constituting
a
sort
of
“arithmetic
holomorphic
structure”
on
a
nonar-
chimedean
G
k
or
an
archimedean
k
×
.
This
arithmetic
holomorphic
structure
is
immune
to
the
log-Frobenius
operation
log
[cf.
the
discussion
of
§I1],
i.e.,
immune
to
the
“juggling
of
,
”
effected
by
log
[cf.
the
illustration
of
Remark
5.10.2,
(iii)].
TOPICS
IN
ABSOLUTE
ANABELIAN
GEOMETRY
III
11
On
the
other
hand,
if
one
exits
such
a
“zone
of
arithmetic
holomorphy”
—
an
operation
that
we
shall
refer
to
as
mono-analyticization
—
then
a
nonar-
chimedean
G
k
or
an
archimedean
k
×
is
stripped
of
the
rigidity
imparted
by
the
above
rigidifying
actions,
hence
may
be
thought
of
as
being
subject
to
Teichmüller
di-
lations
[cf.
Remark
5.10.2,
(ii),
(iii)].
Indeed,
this
is
intuitively
evident
in
the
archimedean
case,
in
which
the
quotient
k
×
R
>0
is
subject
[i.e.,
upon
mono-
analyticization,
so
k
×
is
only
considered
as
a
topological
group]
to
automorphisms
x
→
x
λ
∈
R
>0
,
for
λ
∈
R
>0
.
If,
moreover,
one
thinks
of
of
the
form
R
>0
the
value
groups
of
archimedean
and
nonarchimedean
primes
as
being
“synchro-
nized”
[so
as
to
keep
from
violating
the
product
formula
—
which
plays
a
crucial
role
in
the
theory
of
“heights”,
i.e.,
degrees
of
global
arithmetic
line
bundles],
then
the
operation
of
mono-analyticization
necessarily
results
in
analogous
“Teichmüller
dilations”
at
nonarchimedean
primes.
In
the
context
of
the
theory
of
Frobenioids,
such
Teichmüller
dilations
[whether
archimedean
or
nonarchimedean]
correspond
to
the
unit-linear
Frobenius
functor
studied
in
[Mzk16],
Proposition
2.5.
Note
that
the
“non-linear
juggling
of
,
by
log
within
a
zone
of
arithmetic
holomorphy”
and
the
“linear
Teichmüller
dilations
inherent
in
the
operation
of
mono-analyticization”
are
reminiscent
of
the
Riemannian
geometry
of
the
upper
half-plane,
i.e.,
if
one
thinks
of
“juggling”
as
corresponding
to
rotations
at
a
point,
and
“dilations”
as
corre-
sponding
to
geodesic
flows
originating
from
the
point.
log
mono-analyticization
≈
[linear]
Teichmüller
dilation
---------------------------------------
→
Put
another
way,
the
operation
of
mono-analyticization
may
be
thought
of
as
an
operation
on
the
“arithmetic
holomorphic
structures”
discussed
above
that
forms
a
sort
of
arithmetic
analogue
of
the
operation
of
passing
to
the
underlying
real
analytic
manifold
of
a
Riemann
surface.
number
fields
and
their
localizations
Riemann
surfaces
“arithmetic
holomorphic
structures”
via
rigidifying
hyp.
curves
the
operation
of
mono-analyticization
complex
holomorphic
structure
on
the
Riemann
surface
passing
to
the
underlying
real
analytic
manifold
Thus,
from
this
point
of
view,
one
may
think
of
the
disjoint
union
of
the
various
G
k
’s,
k
×
’s
over
the
various
nonarchimdean
and
archimedean
primes
of
the
number
field
as
being
the
“arithmetic
underlying
real
analytic
manifold”
of
the
“arith-
metic
Riemann
surface”
constituted
by
the
number
field.
Indeed,
it
is
precisely
this
sort
of
disjoint
union
that
arises
in
the
theory
of
mono-analyticization,
as
developed
in
§5.
12
SHINICHI
MOCHIZUKI
Next,
we
consider
the
effect
on
log-shells
of
the
operation
of
mono-analyticization.
In
the
nonarchimedean
case,
log
k
(O
k
×
)
∼
=
O
k
×
/(torsion)
may
be
reconstructed
group-theoretically
from
G
k
as
the
quotient
by
torsion
of
the
image
of
I
k
in
the
abelianization
G
ab
k
[cf.
Proposition
5.8,
(i),
(ii)];
a
similar
con-
struction
may
be
applied
to
finite
extensions
⊆
k
of
k.
Moreover,
this
construction
involves
only
the
group
of
units
O
k
×
[i.e.,
it
does
not
involve
the
value
groups,
which,
as
discussed
above,
are
subject
to
Teichmüller
dilations],
hence
is
compatible
with
the
operation
of
mono-analyticization.
Thus,
this
construction
yields
a
canonical
rigid
integral
structure,
i.e.,
in
the
form
of
the
topological
module
log
k
(O
k
×
),
which
may
be
thought
of
as
a
sort
of
approximation
of
some
nonarchimedean
localization
of
the
trivial
global
arithmetic
line
bundle
[cf.
Remarks
5.4.2,
5.4.3]
that
is
achieved
without
the
use
of
[the
two
combinatorial
dimensions
of]
the
ring
structure
on
O
k
.
Note,
moreover,
that
the
ring
structure
on
the
perfec-
tion
log
k
(O
k
×
)
pf
[i.e.,
in
effect,
“log
k
(O
k
×
)
⊗
Q”]
of
this
module
is
obliterated
by
the
operation
of
mono-analyticization.
That
is
to
say,
this
ring
structure
is
only
acces-
sible
within
a
“zone
of
arithmetic
holomorphy”
[as
discussed
above].
On
the
other
hand,
if
one
returns
to
such
a
zone
of
arithmetic
holomorphy
to
avail
oneself
of
the
ring
structure
on
log
k
(O
k
×
)
pf
,
then
applying
the
operation
of
mono-analyticization
amounts
to
applying
the
construction
discussed
above
to
the
group
of
units
of
log
k
(O
k
×
)
pf
[equipped
with
the
ring
structure
furnished
by
the
zone
of
arithmetic
holomorphy
under
consideration].
That
is
to
say,
the
freedom
to
execute,
at
will,
both
the
operations
of
exiting
and
re-entering
zones
of
arithmetic
holomorphy
is
inextricably
linked
to
the
“juggling
of
,
”
via
log
[cf.
Remark
5.10.2,
(ii),
(iii)].
In
the
archimedean
case,
if
one
writes
log(O
k
×
)
O
k
×
for
the
universal
covering
topological
group
of
O
k
×
[i.e.,
in
essence,
the
exponential
map
“2πi
·
R
S
1
”],
then
the
surjection
log(O
k
×
)
O
k
×
determines
on
log(O
k
×
)
a
“canonical
rigid
line
segment
of
length
2π”.
Thus,
if
one
writes
k
=
k
im
×
k
rl
for
the
product
decomposition
of
the
additive
topological
group
k
into
imaginary
[i.e.,
“i
·
R”]
and
real
[ı.e.,
“R”]
parts,
then
we
obtain
a
natural
isometry
∼
log(O
k
×
)
×
log(O
k
×
)
→
k
im
×
k
rl
=
k
[i.e.,
the
product
of
the
identity
isomorphism
2πi
·
R
=
i
·
R
and
the
isomorphism
∼
2πi
·
R
→
R
given
by
dividing
by
±i]
which
is
well-defined
up
to
multiplication
by
±1
on
the
second
factors
[cf.
Definition
5.6,
(iv);
Proposition
5.8,
(iv),
(v)].
In
particular,
“log(O
k
×
)
×
log(O
k
×
)”
may
be
regarded
as
a
construction,
based
on
the
“rigid”
topological
group
O
k
×
[which
is
not
subject
to
Teichmüller
dilations!],
of
a
canonical
rigid
integral
structure
[determined
by
the
canonical
rigid
line
segments
discussed
above]
that
serves
as
an
approximation
of
some
archimedean
localization
of
the
trivial
global
arithmetic
line
bundle
and,
moreover,
is
compatible
with
the
operation
of
mono-analyticization
[cf.
the
nonarchimedean
case].
On
the
TOPICS
IN
ABSOLUTE
ANABELIAN
GEOMETRY
III
13
other
hand,
[as
might
be
expected
by
comparison
to
the
nonarchimedean
case]
once
one
exits
a
zone
of
arithmetic
holomorphy,
the
±1-indeterminacy
that
occurs
in
the
above
natural
isometry
has
the
effect
of
obstructing
any
attempts
to
transport
the
ring
structure
of
k
via
this
natural
isometry
so
as
to
obtain
a
structure
of
complex
archimedean
field
on
log(O
k
×
)
×
log(O
k
×
)
[cf.
Remark
5.8.1].
Finally,
just
as
in
the
nonarchimedean
case,
the
freedom
to
execute,
at
will,
both
the
operations
of
exiting
and
re-entering
zones
of
arithmetic
holomorphy
is
inextricably
linked
to
the
“juggling
of
,
”
via
log
[cf.
Remark
5.10.2,
(ii),
(iii)]
—
a
phenomenon
that
is
strongly
reminiscent
of
the
crucial
role
played
by
rotations
in
the
theory
of
mono-analyticizations
of
archimedean
log-shells
[cf.
Remark
5.8.1].
§I4.
Mono-anabelian
Log-Frobenius
Compatibility
Within
each
zone
of
arithmetic
holomorphy,
one
wishes
to
apply
the
log-
Frobenius
functor
log.
As
discussed
in
§I1,
log
may
be
thought
of
as
a
sort
of
“wall”
that
may
be
penetrated
by
such
“elementary
combinatorial/topological
ob-
jects”
as
Galois
groups
[in
the
nonarchimedean
case]
or
underlying
topological
spaces
[in
the
archimedean
case],
but
not
by
rings
or
functions
[cf.
Remark
3.7.7].
This
situation
suggests
a
possible
analogy
with
ideas
from
physics
in
which
“étale-like”
structures
[cf.
the
Introduction
of
[Mzk16]],
which
can
penetrate
the
log-wall,
are
regarded
as
“massless”,
like
light,
while
“Frobenius-like”
structures
[cf.
the
Intro-
duction
of
[Mzk16]],
which
cannot
penetrate
the
log-wall,
are
regarded
as
being
of
“positive
mass”,
like
ordinary
matter
[cf.
Remark
3.7.5,
(iii)].
Galois
groups,
∼
=
topological
spaces
-------------------
log
log
Galois
groups,
∼
=
topological
spaces
-------------------
log
rings,
functions
log
rings,
functions
In
the
archimedean
case,
since
topological
spaces
alone
are
not
sufficient
to
trans-
port
“holomorphic
structures”
in
the
usual
sense,
we
take
the
approach
in
§2
of
con-
sidering
“Aut-holomorphic
spaces”,
i.e.,
underlying
topological
spaces
of
Riemann
surfaces
equipped
with
the
additional
data
of
a
group
of
“special
self-homeomor-
phisms”
[i.e.,
bi-holomorphic
automorphisms]
of
each
[sufficiently
small]
open
con-
nected
subset
[cf.
Definition
2.1,
(i)].
The
point
here
is
to
“somehow
encode
the
usual
notion
of
a
holomorphic
structure”
in
such
a
way
that
one
does
not
need
to
resort
to
the
use
of
“fixed
reference
models”
of
the
field
of
complex
numbers
C
[as
is
done
in
the
conventional
definition
of
a
holomorphic
structure,
which
consists
of
local
comparison
to
such
a
fixed
reference
model
of
C],
since
such
models
of
C
fail
to
be
“immune”
to
the
application
of
log
—
cf.
Remarks
2.1.2,
2.7.4.
This
situation
is
very
much
an
archimedean
analogue
of
the
distinction
between
mono-
anabelian
and
bi-anabelian
geometry.
That
is
to
say,
if
one
thinks
of
one
of
the
14
SHINICHI
MOCHIZUKI
two
schemes
that
occur
in
bi-anabelian
comparison
results
as
the
“given
scheme
of
interest”
and
the
other
scheme
as
a
“fixed
reference
model”,
then
although
these
two
schemes
are
related
to
one
another
via
purely
Galois-theoretic
data,
the
scheme
structure
of
the
“scheme
of
interest”
is
reconstructed
from
the
Galois-theoretic
data
by
transporting
the
scheme
structure
of
“model
scheme”,
hence
requires
the
use
of
input
data
[i.e.,
the
scheme
structure
of
the
“model
scheme”]
that
cannot
penetrate
the
log-wall.
In
order
to
formalize
these
ideas
concerning
the
issue
of
distinguishing
between
“model-dependent”,
“bi-anabelian”
approaches
and
“model-independent”,
“mono-
anabelian”
approaches,
we
take
the
point
of
view,
in
§3,
§4,
of
considering
“series
of
operations”
—
in
the
form
of
diagrams
[parametrized
by
various
oriented
graphs]
of
functors
—
applied
to
various
“types
of
data”
—
in
the
form
of
objects
of
categories
[cf.
Remark
3.6.7].
Although,
by
definition,
it
is
impossible
to
compare
the
“different
types
of
data”
obtained
by
applying
these
various
“operations”,
if
one
considers
“projections”
of
these
operations
between
different
types
of
data
onto
morphisms
between
objects
of
a
single
category
[i.e.,
a
single
“type
of
data”],
then
such
comparisons
become
possible.
Such
a
“projection”
is
formalized
in
Definition
3.5,
(iii),
as
the
notion
of
an
observable.
One
special
type
of
observable
that
is
of
crucial
importance
in
the
theory
of
the
present
paper
is
an
observable
that
“captures
a
certain
portion
of
various
distinct
types
of
data
that
remains
constant,
up
to
isomorphism,
throughout
the
series
of
operations
applied
to
these
distinct
types
of
data”.
Such
an
observable
is
referred
to
as
a
core
[cf.
Definition
3.5,
(iii)].
Another
important
notion
in
the
theory
of
the
present
paper
is
the
notion
of
telecore
[cf.
Definition
3.5,
(iv)],
which
may
be
thought
of
as
a
sort
of
“core
structure
whose
compatibility
apparatus
[i.e.,
‘constant
nature’]
only
goes
into
effect
after
a
certain
time
lag”
[cf.
Remark
3.5.1].
Before
explaining
how
these
notions
are
applied
in
the
situation
over
number
fields
considered
in
the
present
paper,
it
is
useful
to
consider
the
analogy
between
these
notions
and
the
classical
p-adic
theory.
The
prototype
of
the
notion
of
a
core
is
the
constant
nature
[i.e.,
up
to
equivalence
of
categories]
of
the
étale
site
of
a
scheme
in
positive
char-
acteristic
with
respect
to
the
[operation
constituted
by
the]
Frobenius
morphism.
Put
another
way,
cores
may
be
thought
of
as
corresponding
to
the
notion
of
“slope
zero”
Galois
representations
in
the
p-adic
theory.
By
contrast,
telecores
may
be
thought
of
as
corresponding
to
the
notion
of
“positive
slope”
in
the
p-adic
theory.
In
particular,
the
“time
lag”
inherent
in
the
compatibility
apparatus
of
a
telecore
may
be
thought
of
as
corresponding
to
the
“lag”,
in
terms
of
powers
of
p,
that
occurs
when
one
applies
Hensel’s
lemma
[cf.,
e.g.,
[Mzk21],
Lemma
2.1]
to
lift
solutions,
modulo
various
powers
of
p,
of
a
polynomial
equation
that
gives
rise
to
a
crystalline
Galois
representation
—
e.g.,
arising
from
an
“MF
∇
-object”
of
[Falt]
—
for
which
the
slopes
of
the
Frobenius
action
are
positive
[cf.
Remark
3.6.5
for
more
on
this
topic].
This
formal
analogy
with
the
classical
p-adic
theory
forms
the
starting
point
for
the
analogy
with
p-adic
Teichmüller
theory
to
be
discussed
in
§I5
below
[cf.
Remark
3.7.2].
TOPICS
IN
ABSOLUTE
ANABELIAN
GEOMETRY
III
15
Now
let
us
return
to
the
situation
involving
k,
k,
G
k
,
and
log
k
discussed
at
the
beginning
of
§I1.
Suppose
further
that
we
are
given
a
hyperbolic
orbicurve
over
k
as
in
the
discussion
of
§I3,
whose
étale
fundamental
group
Π
surjects
onto
G
k
×
[hence
may
be
regarded
as
acting
on
k,
k
]
and,
moreover,
satisfies
the
important
property
of
rigidifying
G
k
[as
discussed
in
§I3].
Then
the
“series
of
operations”
performed
in
this
context
may
be
summarized
as
follows
[cf.
Remark
3.7.3,
(ii)]:
⎞
⎛
Π
⎞
⎟
⎜
⎜
⎟
⎠
⎝
⎜
⎜
⎝
⎟
⎟
⎠
⎛
Π
Π
×
×
k
An
k
⎛
Π
Π
⎞
⎜
⎟
⎟
⎜
⎝
⎠
×
k
An
log
Here,
the
various
operations
“”,
“”
may
be
described
in
words
as
follows:
(O1)
One
applies
the
mono-anabelian
reconstruction
algorithms
of
§1
×
×
×
to
Π
to
construct
a
“mono-anabelian
copy”
k
An
of
k
.
Here,
k
An
is
the
group
of
nonzero
elements
of
a
field
k
An
.
Moreover,
it
is
important
to
note
that
k
An
is
equipped
with
the
structure
not
of
“some
field
k
An
isomorphic
to
k”,
but
rather
of
“the
specific
field
[isomorphic
to
k]
reconstructed
via
the
mono-anabelian
reconstruction
algorithms
of
§1”.
(O2)
One
forgets
the
fact
that
k
An
arises
from
the
mono-anabelian
recon-
struction
algorithms
of
§1,
i.e.,
one
regards
k
An
just
as
“some
field
k
[isomorphic
to
k]”.
(O3)
Having
performed
the
operation
of
(O2),
one
can
now
proceed
to
apply
log-Frobenius
operation
log
[i.e.,
log
k
]
to
k
.
This
operation
log
may
be
thought
of
as
the
assignment
×
(Π
k
)
(Π
{log
k
(O
×
)
pf
}
×
)
k
that
maps
the
group
of
nonzero
elements
of
the
topological
field
k
to
the
group
of
nonzero
elements
of
the
topological
field
“log
k
(O
×
)
pf
”
[cf.
the
k
discussion
of
§I3].
(O4)
One
forgets
all
the
data
except
for
the
profinite
group
Π.
(O5)
This
is
the
same
operation
as
the
operation
described
in
(O1).
With
regard
to
the
operation
log,
observe
that
if
we
forget
the
various
field
or
group
structures
involved,
then
the
arrows
×
k
←
O
×
k
→
log
k
(O
×
)
pf
k
←
{log
k
(O
×
)
pf
}
×
k
allow
one
to
relate
the
input
of
log
[on
the
left]
to
the
the
output
of
log
[on
the
right].
That
is
to
say,
in
the
formalism
developed
in
§3,
these
arrows
may
be
regarded
as
defining
an
observable
“S
log
”
associated
to
log
[cf.
Corollary
3.6,
(iii)].
16
SHINICHI
MOCHIZUKI
If
one
allows
oneself
to
reiterate
the
operation
log,
then
one
obtains
diagrams
equipped
with
a
natural
Z-action
[cf.
Corollary
3.6,
(v)].
These
diagrams
equipped
with
a
Z-action
are
reminiscent,
at
a
combinatorial
level,
of
the
“arithmetic
G
m
’s”
that
occurred
in
the
discussion
of
§I3
[cf.
Remark
3.6.3].
Next,
observe
that
the
operation
of
“projecting
to
Π”
[i.e.,
forgetting
all
of
the
data
under
consideration
except
for
Π]
is
compatible
with
the
execution
of
any
of
these
operations
(O1),
(O2),
(O3),
(O4),
(O5).
That
is
to
say,
Π
determines
a
core
of
this
collection
of
operations
[cf.
Corollary
3.6,
(i),
(ii),
(iii)].
Moreover,
since
the
mono-anabelian
reconstruction
algorithms
of
§1
are
“purely
group-theoretic”
and
depend
only
on
the
input
data
constituted
by
Π,
it
follows
immediately
that
[by
×
“projecting
to
Π”
and
then
applying
these
algorithms]
“(Π
k
An
)”
also
forms
a
core
of
this
collection
of
operations
[cf.
Corollary
3.6,
(i),
(ii),
(iii)].
In
particular,
×
we
obtain
a
natural
isomorphism
between
the
“(Π
k
An
)’s”
that
occur
following
the
first
and
fourth
“’s”
of
the
above
diagram.
On
the
other
hand,
the
“forgetting”
operation
of
(O2)
may
be
thought
of
as
a
×
sort
of
section
of
the
“projection
to
the
core
(Π
k
An
)”.
This
sort
of
section
will
be
referred
to
as
a
telecore;
a
telecore
frequently
comes
equipped
with
an
auxiliary
structure,
called
a
contact
structure,
which
corresponds
in
the
present
situation
to
the
isomorphism
of
underlying
fields
[stripped
of
their
respective
zero
elements]
×
∼
×
×
k
An
→
k
[cf.
Corollary
3.6,
(ii)].
Even
though
the
core
“(Π
k
An
)”,
regarded
as
an
object
obtained
by
projecting,
is
constant
[up
to
isomorphism],
the
section
obtained
in
this
way
does
not
yield
a
“constant”
collection
of
data
[with
respect
to
the
operations
of
the
diagram
above]
that
is
compatible
with
the
observable
S
log
.
×
Indeed,
forgetting
the
marker
“An”
of
[the
constant]
k
An
and
then
applying
log
is
not
compatible,
relative
to
S
log
,
with
forgetting
the
marker
“An”
—
i.e.,
since
log
obliterates
the
ring
structures
involved
[cf.
Corollary
3.6,
(iv);
Remark
3.6.1].
Nevertheless,
if,
subsequent
to
applying
the
operations
of
(O2),
(O3),
one
×
projects
back
down
to
“(Π
k
An
)”,
then,
as
was
observed
above,
one
obtains
a
×
natural
isomorphism
between
the
initial
and
final
copies
of
“(Π
k
An
)”.
It
is
in
this
sense
that
one
may
think
of
a
telecore
as
a
“core
with
a
time
lag”.
One
way
to
summarize
the
above
discussion
is
as
follows:
The
“purely
group-
theoretic”
mono-anabelian
reconstruction
algorithms
of
§1
allow
one
to
construct
×
models
of
scheme-theoretic
data
[i.e.,
the
“k
An
”]
that
satisfy
the
following
three
properties
[cf.
Remark
3.7.3,
(i),
(ii)]:
×
(P1)
coricity
[i.e.,
the
“property
of
being
a
core”
of
“Π”,
“(Π
k
An
)”];
(P2)
comparability
[i.e.,
via
the
telecore
and
contact
structures
discussed
×
above]
with
log-subject
copies
[i.e.,
the
“k
”,
which
are
subject
to
the
action
of
log];
(P3)
log-observability
[i.e.,
via
“S
log
”].
One
way
to
understand
better
what
is
gained
by
this
mono-anabelian
approach
is
to
consider
what
happens
if
one
takes
a
bi-anabelian
approach
to
this
situation
[cf.
Remarks
3.7.3,
(iii),
(iv);
3.7.5].
TOPICS
IN
ABSOLUTE
ANABELIAN
GEOMETRY
III
17
In
the
bi-anabelian
approach,
instead
of
taking
just
“Π”
as
one’s
core,
one
takes
the
data
×
(Π
k
model
)
—
where
“k
model
”
is
some
fixed
reference
model
of
k
—
as
one’s
core
[cf.
Corollary
3.7,
(i)].
The
bi-anabelian
version
[i.e.,
fully
faithfulness
in
the
style
of
the
“Grothendieck
Conjecture”]
of
the
mono-anabelian
theory
of
§1
then
gives
rise
×
×
∼
to
telecore
and
contact
structures
by
considering
the
isomorphism
k
model
→
k
×
×
arising
from
an
isomorphism
between
the
“Π’s”
that
act
on
k
model
,
k
[cf.
Corollary
3.7,
(ii)].
Moreover,
one
may
define
an
observable
“S
log
”
as
in
the
mono-anabelian
case
[cf.
Corollary
3.7,
(iii)].
Just
as
in
the
mono-anabelian
case,
since
log
obliterates
×
the
ring
structures
involved,
this
model
k
model
fails
to
be
simultaneously
compatible
with
the
observable
S
log
and
the
telecore
and
[a
slight
extension,
as
described
in
Corollary
3.7,
(ii),
of
the]
contact
structures
just
mentioned
[cf.
Corollary
3.7,
(iv)].
On
the
other
hand,
whereas
in
the
mono-anabelian
case,
one
may
recover
from
this
failure
of
compatibility
by
projecting
back
down
to
“Π”
[which
remains
×
×
intact!]
and
hence
to
“(Π
k
An
)”,
in
the
bi-anabelian
case,
the
“k
model
”
portion
×
of
“the
core
(Π
k
model
)”
—
which
is
an
essential
portion
of
the
input
data
for
reconstruction
algorithms
via
the
bi-anabelian
approach!
[cf.
Remarks
3.7.3,
(iv);
3.7.5,
(ii)]
—
is
obliterated
by
log,
thus
rendering
it
impossible
to
relate
the
×
“(Π
k
model
)’s”
before
and
after
the
application
of
log
via
an
isomorphism
that
is
compatible
with
all
of
the
operations
involved.
At
a
more
technical
level,
the
non-
existence
of
such
a
natural
isomorphism
may
be
seen
in
the
fact
that
the
coricity
of
×
“(Π
k
model
)”
is
only
asserted
in
Corollary
3.7,
(i),
for
a
certain
limited
portion
of
the
diagram
involving
“all
of
the
operations
under
consideration”
[cf.
also
the
incompatibilities
of
Corollary
3.7,
(iv)].
This
contrasts
with
the
[manifest!]
coricity
×
of
“Π”,
“(Π
k
An
)”
with
respect
to
all
of
the
operations
under
consideration
in
the
mono-anabelian
case
[cf.
Corollary
3.6,
(i),
(ii),
(iii)].
In
this
context,
one
important
observation
is
that
if
one
tries
to
“subsume”
the
×
model
“k
model
”
into
Π
by
“regarding”
this
model
as
an
object
that
“arises
from
the
sole
input
data
Π”,
then
one
must
contend
with
various
problems
from
the
point
of
view
of
functoriality
—
cf.
Remark
3.7.4
for
more
details
on
such
“functorially
×
trivial
models”.
That
is
to
say,
to
regard
“k
model
”
in
this
way
means
that
one
must
contend
with
a
situation
in
which
the
functorially
induced
action
of
Π
on
×
“k
model
”
is
trivial!
Finally,
we
note
in
passing
that
the
“dynamics”
of
the
various
diagrams
of
operations
[i.e.,
functors]
appearing
in
the
above
discussion
are
reminiscent
of
the
analogy
with
physics
discussed
at
the
beginning
of
the
present
§I4
—
i.e.,
that
“Π”
×
is
massless,
like
light,
while
“k
”
is
of
positive
mass.
§I5.
Analogy
with
p-adic
Teichmüller
Theory
We
have
already
discussed
in
§I1
the
analogy
between
the
log-Frobenius
oper-
ation
log
and
the
Frobenius
morphism
in
positive
characteristic.
This
analogy
may
be
developed
further
[cf.
Remarks
3.6.6,
3.7.2
for
more
details]
into
an
analogy
be-
tween
the
formalism
discussed
in
§I4
and
the
notion
of
a
uniformizing
MF
∇
-object
18
SHINICHI
MOCHIZUKI
as
discussed
in
[Mzk1],
[Mzk4],
i.e.,
an
MF
∇
-object
in
the
sense
of
[Falt]
that
gives
rise
to
“canonical
coordinates”
that
may
be
regarded
as
a
sort
of
p-adic
uni-
formization
of
the
variety
under
consideration.
Indeed,
in
the
notation
of
§I4,
the
×
“mono-anabelian
output
data
(Π
k
An
)”
may
be
regarded
as
corresponding
to
the
“Galois
representation”
associated
to
a
structure
of
“uniformizing
MF
∇
-
×
object”
on
the
scheme-theoretic
“(Π
k
)”.
The
telecore
structures
discussed
in
§I4
may
be
regarded
as
corresponding
to
a
sort
of
Hodge
filtration,
i.e.,
an
opera-
tion
relating
the
“Frobenius
crystal”
under
consideration
to
a
specific
scheme
theory
×
“(Π
k
)”,
among
the
various
scheme
theories
separated
from
one
another
by
[the
non-ring-homomorphism!]
log.
The
associated
contact
structures
then
take
on
an
appearance
that
is
formally
reminiscent
of
the
notion
of
a
connection
in
the
classical
crystalline
theory.
The
failure
of
the
log-observable,
telecore,
and
contact
structures
to
be
simultaneously
compatible
[cf.
Corollaries
3.6,
(iv);
3.7,
(iv)]
may
then
be
regarded
as
corresponding
to
the
fact
that,
for
instance
in
the
case
of
the
uniformizing
MF
∇
-objects
determined
by
indigenous
bundles
in
[Mzk1],
[Mzk4],
the
Kodaira-Spencer
morphism
is
an
isomorphism
[i.e.,
the
fact
that
the
Hodge
filtration
fails
to
be
a
horizontal
Frobenius-invariant!].
mono-anabelian
theory
p-adic
theory
log-Frobenius
log
mono-anabelian
output
data
telecore
structure
contact
structure
simultaneous
incompatibility
of
log-observable,
telecore,
and
contact
structures
Frobenius
Frobenius-invariants
Hodge
filtration
connection
Kodaira-Spencer
morphism
of
an
indigenous
bundle
is
an
isomorphism
In
the
context
of
this
analogy,
we
observe
that
the
failure
of
the
logarithms
at
the
various
localizations
of
a
number
field
to
extend
to
a
global
map
involving
the
number
field
[cf.
Remark
5.4.1]
may
be
regarded
as
corresponding
to
the
fail-
ure
of
various
Frobenius
liftings
on
affine
opens
[i.e.,
localizations]
of
a
hyperbolic
curve
[over,
say,
a
ring
of
Witt
vectors
of
a
perfect
field]
to
extend
to
a
morphism
defined
[“globally”]
on
the
entire
curve
[cf.
[Mzk21],
Remark
2.6.2].
This
lack
of
a
global
extension
in
the
p-adic
case
means,
in
particular,
that
it
does
not
make
sense
to
pull-back
arbitrary
coherent
sheaves
on
the
curve
via
such
Frobenius
lift-
ings.
On
the
other
hand,
if
a
coherent
sheaf
on
the
curve
is
equipped
with
the
structure
of
a
crystal,
then
a
“global
pull-back
of
the
crystal”
is
well-defined
and
“canonical”,
even
though
the
various
local
Frobenius
liftings
used
to
construct
it
are
not.
In
a
similar
way,
although
the
logarithms
at
localizations
of
a
number
field
are
not
compatible
with
the
ring
structures
involved,
hence
cannot
be
used
to
pull-back
arbitrary
ring/scheme-theoretic
objects,
they
can
be
used
to
“pull-back”
Galois-theoretic
structures,
such
as
those
obtained
by
applying
mono-anabelian
re-
construction
algorithms.
TOPICS
IN
ABSOLUTE
ANABELIAN
GEOMETRY
III
mono-anabelian
theory
p-adic
theory
logarithms
at
localizations
of
a
number
field
nonexistence
of
global
logarithm
on
a
number
field
incompatibility
of
log
with
ring
structures
compatibility
of
log
with
Galois,
mono-anabelian
algorithms
the
result
of
forgetting
“An”
[cf.
(O2)
of
§I3]
Frobenius
liftings
on
affine
opens
of
a
hyperbolic
curve
nonexistence
of
global
Frobenius
lifting
on
a
hyperbolic
curve
noncanonicality
of
local
liftings
of
positive
characteristic
Frobenius
Frobenius
pull-back
of
crystals
the
underlying
coherent
sheaf
of
a
crystal
19
Moreover,
this
analogy
may
be
developed
even
further
by
specializing
from
arbitrary
uniformizing
MF
∇
-objects
to
the
indigenous
bundles
of
the
p-adic
Te-
ichmüller
theory
of
[Mzk1],
[Mzk4].
To
see
this,
we
begin
by
observing
that
the
non-rigid
dimension
of
the
localizations
of
a
number
field
“G
k
”,
“k
×
”
in
the
discus-
sion
of
§I3
may
be
regarded
as
analogous
to
the
non-rigidity
of
a
p-adic
deformation
of
an
affine
open
[i.e.,
a
localization]
of
a
hyperbolic
curve
in
positive
character-
istic.
If,
on
the
other
hand,
such
a
hyperbolic
curve
is
equipped
with
the
crystal
determined
by
a
p-adic
indigenous
bundle,
then,
even
if
one
restricts
to
an
affine
open,
this
filtered
crystal
has
the
effect
of
rigidifying
a
specific
p-adic
deformation
of
this
affine
open.
Indeed,
this
rigidifying
effect
is
an
immediate
consequence
of
the
fact
that
the
Kodaira-Spencer
morphism
of
an
indigenous
bundle
is
an
iso-
morphism.
Put
another
way,
this
Kodaira-Spencer
isomorphism
has
the
effect
of
allowing
the
affine
open
to
“entrust
its
moduli”
to
the
crystal
determined
by
the
p-adic
indigenous
bundle.
This
situation
is
reminiscent
of
the
rigidifying
ac-
tions
discussed
in
§I3
of
“G
k
”,
“k
×
”
on
certain
geometric
data
arising
from
a
hyperbolic
orbicurve
that
is
related
to
a
once-punctured
elliptic
curve.
That
is
to
say,
the
mono-anabelian
theory
of
§1,
§2
allows
these
localizations
“G
k
”,
“k
×
”
of
a
number
field
to
“entrust
their
ring
structures”
—
i.e.,
their
“arithmetic
holomorphic
moduli”
—
to
the
hyperbolic
orbicurve
under
consideration.
This
leads
naturally
[cf.
Remark
5.10.3,
(i)]
to
the
analogy
already
referred
to
in
§I1:
mono-anabelian
theory
p-adic
theory
number
field
F
once-punctured
ell.
curve
X
over
F
hyperbolic
curve
C
in
pos.
char.
nilp.
ord.
indig.
bundle
P
over
C
If,
moreover,
one
modifies
the
canonical
rigid
integral
structures
furnished
by
log-
shells
by
means
of
the
“Gaussian
zeroes”
[i.e.,
the
inverse
of
the
“Gaussian
poles”]
that
appear
in
the
Hodge-Arakelov
theory
of
elliptic
curves
[cf.,
e.g.,
[Mzk6],
§1.1],
then
one
may
further
refine
the
above
analogy
by
regarding
indigenous
bun-
dles
as
corresponding
to
the
crystalline
theta
object
[which
may
be
thought
of
as
20
SHINICHI
MOCHIZUKI
an
object
obtained
by
equipping
a
direct
sum
of
trivial
line
bundles
with
the
inte-
gral
structures
determined
by
the
Gaussian
zeroes]
of
Hodge-Arakelov
theory
[cf.
Remark
5.10.3,
(ii)].
From
this
point
of
view,
the
mono-anabelian
theory
of
§1,
§2,
which
may
be
thought
of
as
centering
around
the
technique
of
Belyi
cuspidal-
izations,
may
be
regarded
as
corresponding
to
the
theory
of
indigenous
bundles
in
positive
characteristic
[cf.
[Mzk1],
Chapter
II],
which
centers
around
the
Ver-
schiebung
on
indigenous
bundles.
Moreover,
the
theory
of
the
étale
theta
function
given
in
[Mzk18],
which
centers
around
the
technique
of
elliptic
cuspidalizations,
may
be
regarded
as
corresponding
to
the
theory
of
the
Frobenius
action
on
square
differentials
in
[Mzk1],
Chapter
II.
Indeed,
just
as
the
technique
of
elliptic
cuspidal-
izations
may
be
thought
of
a
sort
of
linearized,
simplified
version
of
the
technique
of
Belyi
cuspidalizations,
the
Frobenius
action
on
square
differentials
occurs
as
the
derivative
[i.e.,
a
“linearized,
simplified
version”]
of
the
Verschiebung
on
indige-
nous
bundles.
For
more
on
this
analogy,
we
refer
to
Remark
5.10.3.
In
passing,
we
observe,
relative
to
the
point
of
view
that
the
theory
of
the
étale
theta
func-
tion
given
in
[Mzk18]
somehow
represents
a
“linearized,
simplified
version”
of
the
mono-anabelian
theory
of
the
present
paper,
that
the
issue
of
mono-
versus
bi-
anabelian
geometry
discussed
in
the
present
paper
is
vaguely
reminiscent
of
the
issue
of
mono-
versus
bi-theta
environments,
which
constitutes
a
central
theme
in
[Mzk18].
In
this
context,
it
is
perhaps
natural
to
regard
the
“log-wall”
discussed
in
§I4
—
which
forms
the
principal
obstruction
to
applying
the
bi-anabelian
approach
in
the
present
paper
—
as
corresponding
to
the
“Θ-wall”
constituted
by
the
theta
function
between
the
theta
and
algebraic
trivializations
of
a
certain
ample
line
bun-
dle
—
which
forms
the
principal
obstruction
to
the
use
of
bi-theta
environments
in
the
theory
of
[Mzk18].
mono-anabelian
theory
p-adic
theory
crystalline
theta
objects
in
scheme-theoretic
Hodge-Arakelov
theory
Belyi
cuspidalizations
in
mono-anabelian
theory
of
§1
elliptic
cuspidalizations
in
the
theory
of
the
étale
theta
function
[cf.
[Mzk18]]
scheme-theoretic
indigenous
bundles
[cf.
[Mzk1],
Chapter
I]
Verschiebung
on
pos.
char.
indigenous
bundles
[cf.
[Mzk1],
Chapter
II]
Frobenius
action
on
square
differentials
[cf.
[Mzk1],
Chapter
II]
Thus,
in
summary,
the
analogy
discussed
above
may
be
regarded
as
an
anal-
ogy
between
the
theory
of
the
present
paper
and
the
positive
characteristic
portion
of
the
theory
of
[Mzk1].
This
“positive
characteristic
portion”
may
be
regarded
as
including,
in
a
certain
sense,
the
“liftings
modulo
p
2
portion”
of
the
theory
of
[Mzk1]
since
this
“liftings
modulo
p
2
portion”
may
be
formulated,
to
a
certain
extent,
in
terms
of
positive
characteristic
scheme
theory.
If,
moreover,
one
re-
gards
the
theory
of
mono-anabelian
log-Frobenius
compatibility
as
corresponding
to
“Frobenius
liftings
modulo
p
2
”,
then
the
isomorphism
between
Galois
groups
on
both
sides
of
the
log-wall
may
be
thought
of
as
corresponding
to
the
Frobenius
TOPICS
IN
ABSOLUTE
ANABELIAN
GEOMETRY
III
21
action
on
differentials
induced
by
dividing
the
derivative
of
such
a
Frobenius
lift-
ing
modulo
p
2
by
p.
This
correspondence
between
Galois
groups
and
differentials
is
reminiscent
of
the
discussion
in
[Mzk6],
§1.3,
§1.4,
of
the
arithmetic
Kodaira-
Spencer
morphism
that
arises
from
the
[scheme-theoretic]
Hodge-Arakelov
theory
of
elliptic
curves.
Finally,
from
this
point
of
view,
it
is
perhaps
natural
to
regard
the
mono-anabelian
reconstruction
algorithms
of
§1
as
corresponding
to
the
procedure
of
integrating
Frobenius-invariant
differentials
so
as
to
obtain
canonical
coordinates
[i.e.,
“q-parameters”
—
cf.
[Mzk1],
Chapter
III,
§1].
mono-anabelian
theory
p-adic
theory
isomorphism
between
Galois
groups
on
both
sides
of
log-wall
mono-anabelian
reconstruction
algorithms
Frobenius
action
on
differentials
arising
from
p
1
·
derivative
of
mod
p
2
Frobenius
lifting
construction
of
can.
coords.
via
integration
of
Frobenius-invariant
differentials
The
above
discussion
prompts
the
following
question:
Can
one
further
extend
the
theory
given
in
the
present
paper
to
a
theory
that
is
analogous
to
the
theory
of
canonical
p-adic
liftings
given
in
[Mzk1],
Chapter
III?
It
is
the
intention
of
the
author
to
pursue
the
goal
of
developing
such
an
“extended
theory”
in
a
future
paper.
Before
proceeding,
we
note
that
the
analogy
of
such
a
theory
with
the
theory
of
canonical
p-adic
liftings
of
[Mzk1],
Chapter
III,
may
be
thought
of
as
a
sort
of
p-adic
analogue
of
the
“geodesic
flow”
portion
of
the
“rotations
and
geodesic
flows
diagram”
of
§I3:
mono-anabelian
theory
p-adic
theory
mono-anabelian
juggling
of
present
paper,
i.e.,
“rotations”
future
extended
theory
(?),
i.e.,
“geodesic
flows”
positive
characteristic
[plus
mod
p
2
]
portion
of
p-adic
Teichmüller
theory
canonical
p-adic
liftings
in
p-adic
Teichmüller
theory
—
that
is
to
say,
p-adic
deformations
correspond
to
“geodesic
flows”,
while
the
positive
characteristic
theory
corresponds
to
“rotations”
[i.e.,
the
theory
of
“mono-
anabelian
juggling
of
,
via
log”
given
in
the
present
paper].
This
point
of
view
is
reminiscent
of
the
analogy
between
the
archimedean
and
nonarchimedean
theories
discussed
in
Table
1
of
the
Introduction
to
[Mzk14].
In
this
context,
it
is
interesting
to
note
that
this
analogy
between
the
mono-
anabelian
theory
of
the
present
paper
and
p-adic
Teichmüller
theory
is
reminiscent
of
various
phenomena
that
appear
in
earlier
papers
by
the
author:
22
SHINICHI
MOCHIZUKI
(A1)
In
[Mzk10],
Theorem
3.6,
an
absolute
p-adic
anabelian
result
is
obtained
for
canonical
curves
as
in
[Mzk1]
by
applying
the
p-adic
Teichmüller
theory
of
[Mzk1].
Thus,
in
a
certain
sense
[i.e.,
“Teichmüller
=⇒
anabelian”
as
opposed
to
“anabelian
=⇒
Teichmüller”],
this
result
goes
in
the
opposite
direction
to
the
direction
of
the
theory
of
the
present
paper.
On
the
other
hand,
this
result
of
[Mzk10]
depends
on
the
analysis
in
[Mzk9],
§2,
of
the
logarithmic
special
fiber
of
a
p-adic
hyperbolic
curve
via
absolute
anabelian
geometry
over
finite
fields.
(A2)
The
reconstruction
of
the
“additive
structure”
via
the
mono-anabelian
algorithms
of
§1
[cf.
the
lemma
of
Uchida
reviewed
in
Proposition
1.3],
which
eventually
leads
[as
discussed
above],
via
the
theory
of
§3,
to
an
abstract
analogue
of
“Frobenius
liftings”
[i.e.,
in
the
form
of
uniformizing
MF
∇
-objects]
is
reminiscent
[cf.
Remark
5.10.4]
of
the
reconstruction
of
the
“additive
structure”
in
[Mzk21],
Corollary
2.9,
via
an
argument
analogous
to
an
argument
that
may
be
used
to
show
the
non-existence
of
Frobenius
liftings
on
p-adic
hyperbolic
curves
[cf.
[Mzk21],
Remark
2.9.1].
One
way
to
think
about
(A1),
(A2)
is
by
considering
the
following
chart:
p-adic
Teichmüller
Theory
(applied
to
anabelian
geometry)
Uchida’s
Lemma
applied
to:
characteristic
p
special
fiber
Deformation
Theory:
canonical
p-adic
Frobenius
liftings
Future
“Teichmüller-like”
Extension
(?)
of
Mono-anabelian
Theory
number
fields,
mixed
char.
local
fields
equipped
with
an
elliptically
admissible
hyperbolic
orbicurve
analogue
of
Frobenius
liftings
in
future
extension
(?)
of
mono-anabelian
theory
Here,
the
correspondence
in
the
first
non-italicized
line
between
hyperbolic
curves
in
positive
characteristic
equipped
with
a
nilpotent
ordinary
indigenous
bundle
and
number
fields
[and
their
localizations]
equipped
with
an
elliptically
admissible
hy-
perbolic
orbicurve
[i.e.,
a
hyperbolic
orbicurve
closely
related
to
a
once-punctured
elliptic
curve]
has
already
been
discussed
above;
the
content
of
the
“p-adic
Te-
ichmüller
theory
column”
of
this
chart
may
be
thought
of
as
a
summary
of
the
content
of
(A1);
the
correspondence
between
this
column
and
the
“extended
mono-
anabelian
theory”
column
may
be
regarded
as
a
summary
of
the
preceding
dis-
cussion.
On
the
other
hand,
the
content
of
(A2)
may
be
thought
of
as
a
sort
of
“remarkable
bridge”
canonical
p-adic
Frobenius
liftings
theory
of
geometric
uniformly
toral
neighborhoods
===========
number
fields,
mixed
char.
local
fields
equipped
with
an
elliptically
admissible
hyperbolic
orbicurve
TOPICS
IN
ABSOLUTE
ANABELIAN
GEOMETRY
III
23
between
the
upper
right-hand
and
lower
left-hand
non-italicized
entries
of
the
above
chart.
That
is
to
say,
the
theory
of
(A2)
[i.e.,
of
geometric
uniformly
toral
neigh-
borhoods
—
cf.
[Mzk21],
§2]
is
related
to
the
upper
right-hand
non-italicized
entry
of
the
chart
in
that,
like
the
application
of
“Uchida’s
Lemma”
represented
by
this
entry,
it
provides
a
means
for
recovering
the
ring
structure
of
the
base
field,
given
the
decomposition
groups
of
the
closed
points
of
the
hyperbolic
orbicurve.
On
the
other
hand,
the
theory
of
(A2)
[i.e.,
of
[Mzk21],
§2]
is
related
to
the
lower
left-hand
entry
of
the
chart
in
that
the
main
result
of
[Mzk21],
§2,
is
obtained
by
an
argument
reminiscent
[cf.
[Mzk21],
Remark
2.6.2;
[Mzk21],
Remark
2.9.1]
of
the
argument
to
the
effect
that
stable
curves
over
rings
of
Witt
vectors
of
a
perfect
field
never
admit
Frobenius
liftings.
Note,
moreover,
that
from
the
point
of
view
of
the
discussion
above
of
“arith-
metic
holomorphic
structures”,
this
bridge
may
be
thought
of
as
a
link
between
the
elementary
algebraic
approach
to
reconstructing
the
“two
combinatorial
dimen-
sions”
of
a
ring
in
the
fashion
of
Uchida’s
Lemma
and
the
“p-adic
differential-
geometric
approach”
to
reconstructing
p-adic
ring
structures
in
the
fashion
of
the
theory
of
[Mzk21],
§2.
Here,
we
observe
that
this
“p-adic
differential-geometric
approach”
makes
essential
use
of
the
hyperbolicity
of
the
curve
under
consideration.
Indeed,
roughly
speaking,
from
the
“Teichmüller-theoretic”
point
of
view
of
the
present
discussion,
the
argument
of
the
proof
of
[Mzk21],
Lemma
2.6,
(ii),
may
be
summarized
as
follows:
The
nonexistence
of
the
desired
“geometric
uniformly
toral
neighborhoods”
may
be
thought
of
as
a
sort
of
nonexistence
of
obstructions
to
Teichmüller
deformations
of
the
“arithmetic
holomorphic
structure”
that
extend
in
an
unbounded,
linear
fashion,
like
a
geodesic
flow
or
Frobenius
lifting.
On
the
other
hand,
the
hyperbolicity
of
the
curve
under
consideration
implies
the
existence
of
topological
obstructions
—
i.e.,
in
the
form
of
“loopifica-
tion”
or
“crushed
components”
[cf.
[Mzk21],
Lemma
2.6,
(ii)]
—
to
such
“unbounded”
deformations
of
the
holomorphic
structure.
Moreover,
such
“compact
bounds”
on
the
deformability
of
the
holomorphic
structure
are
sufficient
to
“trap”
the
holomorphic
structure
at
a
“canonical
point”,
which
corresponds
to
the
original
holomorphic
[i.e.,
ring]
structure
of
interest.
Put
another
way,
this
“p-adic
differential-geometric
interpretation
of
hyperbolicity”
is
reminiscent
of
the
dynamics
of
a
rubber
band,
whose
elasticity
implies
that
even
if
one
tries
to
stretch
the
rubber
band
in
an
unbounded
fashion,
the
rubber
band
ultimately
returns
to
a
“canonical
position”.
Moreover,
this
relationship
be-
tween
hyperbolicity
and
“elasticity”
is
reminiscent
of
the
use
of
the
term
“elastic”
in
describing
certain
group-theoretic
aspects
of
hyperbolicity
in
the
theory
of
[Mzk20],
§1,
§2.
In
passing,
we
observe
that
another
important
aspect
of
the
theory
of
[Mzk21],
§2,
in
the
present
context
is
the
use
of
the
inequality
of
degrees
obtained
by
“differentiating
a
Frobenius
lifting”
[cf.
[Mzk21],
Remark
2.6.2].
The
key
importance
of
such
degree
inequalities
in
the
theory
of
[Mzk21],
§2,
suggests,
relative
to
the
above
chart,
that
the
analogue
of
such
degree
inequalities
in
the
theory
of
“the
analogue
of
Frobenius
liftings
in
a
future
extension
of
the
mono-anabelian
24
SHINICHI
MOCHIZUKI
theory”
could
give
rise
to
results
of
substantial
interest
in
the
arithmetic
of
number
fields.
The
author
hopes
to
address
this
topic
in
more
detail
in
a
future
paper.
Finally,
we
close
the
present
Introduction
to
the
present
paper
with
some
historical
remarks.
We
begin
by
considering
the
following
historical
facts:
(H1)
O.
Teichmüller,
in
his
relatively
short
career
as
a
mathematician,
made
contributions
both
to
“complex
Teichmüller
theory”
and
to
the
theory
of
Teichmüller
representatives
of
Witt
rings
—
two
subjects
that,
at
first
glance,
appear
entirely
unrelated
to
one
another.
(H2)
In
the
Introduction
to
[Ih],
Y.
Ihara
considers
the
issue
of
obtaining
canonical
p-adic
liftings
of
certain
positive
characteristic
hyperbolic
curves
equipped
with
a
correspondence
in
a
fashion
analogous
to
the
Serre-Tate
theory
of
canonical
liftings
of
abelian
varieties.
These
two
facts
may
be
regarded
as
interesting
precursors
of
the
p-adic
Teichmüller
theory
of
[Mzk1],
[Mzk4].
Indeed,
the
p-adic
Teichmüller
theory
of
[Mzk1],
[Mzk4]
may
be
regarded,
on
the
one
hand,
as
an
analogue
for
hyperbolic
curves
of
the
Serre-Tate
theory
of
canonical
liftings
of
abelian
varieties
and,
on
the
other
hand,
as
a
p-adic
analogue
of
complex
Teichmüller
theory;
moreover,
the
canonical
liftings
obtained
in
[Mzk1],
[Mzk4]
are,
literally,
“hyperbolic
curve
versions
of
Teichmüller
representatives
in
Witt
rings”.
In
fact,
one
may
even
go
one
step
further
to
speculate
that
perhaps
the
existence
of
analogous
complex
and
p-adic
versions
of
“Teichmüller
theory”
should
be
regarded
as
hinting
of
a
deeper
abstract,
combinatorial
version
of
“Teichmüller
theory”
—
in
a
fashion
that
is
perhaps
reminiscent
of
the
relationship
of
the
notion
of
a
motive
to
various
complex
or
p-adic
cohomology
theories.
It
is
the
hope
of
the
author
that
a
possible
“future
extended
theory”
as
discussed
above
—
i.e.,
a
sort
of
“Teichmüller
theory”
for
number
fields
equipped
with
a
once-punctured
elliptic
curve
—
might
prove
to
be
just
such
a
“Teichmüller
theory”.
Acknowledgements:
I
would
like
to
thank
Yuichiro
Hoshi,
Fumiharu
Kato,
Akio
Tamagawa,
and
Go
Yamashita
for
countless
stimulating
discussions
concerning
the
material
presented
in
this
paper
and
Minhyong
Kim
and
Ben
Moonen
for
stimulating
questions
related
to
certain
of
the
ideas
presented
in
this
paper.
Also,
I
feel
most
indebted
to
Yuichiro
Hoshi,
Go
Yamashita,
and
Mohamed
Saı̈di
for
their
meticulous
reading
of
and
numerous
comments
concerning
the
present
paper.
Finally,
I
would
like
to
thank
Ivan
Fesenko
for
his
comments
concerning
the
present
paper.
TOPICS
IN
ABSOLUTE
ANABELIAN
GEOMETRY
III
25
Section
0:
Notations
and
Conventions
We
shall
continue
to
use
the
“Notations
and
Conventions”
of
[Mzk20],
§0;
[Mzk21],
§0.
In
addition,
we
shall
use
the
following
notation
and
conventions:
Numbers:
In
addition
to
the
“field
types”
NF,
MLF,
FF
introduced
in
[Mzk20],
§0,
we
shall
also
consider
the
following
field
types:
A(n)
complex
archimedean
field
(re-
spectively,
real
archimedean
field;
archimedean
field),
or
CAF
(respectively,
RAF;
AF),
is
defined
to
be
a
topological
field
that
is
isomorphic
to
the
field
of
complex
numbers
(respectively,
the
field
of
real
numbers;
either
the
field
of
real
numbers
or
the
field
of
complex
numbers).
One
verifies
immediately
that
any
continuous
homomorphism
between
CAF’s
(respectively,
RAF’s)
is,
in
fact,
an
isomorphism
of
topological
fields.
Combinatorics:
Let
E
be
a
partially
ordered
set.
Then
[cf.
[Mzk16],
§0]
we
shall
denote
by
Order(E)
the
category
whose
objects
are
elements
e
∈
E,
and
whose
morphisms
e
1
→
e
2
[where
e
1
,
e
2
∈
E]
are
the
relations
e
1
≤
e
2
.
A
subset
E
⊆
E
will
be
called
orderwise
connected
if
for
every
c
∈
E
such
a
<
c
<
b
for
some
a,
b
∈
E
,
it
follows
that
c
∈
E
.
A
partially
ordered
set
which
is
isomorphic
[as
a
partially
ordered
set]
to
an
orderwise
connected
subset
of
the
set
of
rational
integers
Z,
equipped
with
its
usual
ordering,
will
be
referred
to
as
a
countably
ordered
set.
If
E
is
a
countably
ordered
set,
then
any
choice
of
an
isomorphism
of
E
with
an
orderwise
connected
subset
E
⊆
Z
allows
one
to
define
[in
a
fashion
independent
of
the
choice
of
E
],
for
non-maximal
(respectively,
non-minimal)
e
∈
E
[i.e.,
e
such
that
there
exists
an
f
∈
E
that
is
>
e
(respectively,
<
e)],
an
element
“e
+
1”
(respectively,
“e
−
1”)
of
E.
Pairs
of
elements
of
E
of
the
form
(e,
e
+
1)
will
be
referred
to
as
adjacent.
An
oriented
graph
Γ
is
a
graph
Γ,
which
we
shall
refer
to
as
the
underlying
graph
of
Γ,
equipped
with
the
additional
data
of
a
total
ordering,
for
each
edge
e
of
Γ,
on
the
set
[of
cardinality
2]
of
branches
of
e
[cf.,
e.g.,
[Mzk13],
the
discussion
at
the
beginning
of
§1,
for
a
definition
of
the
terms
“graph”,
“branch”].
In
this
situation,
we
shall
refer
to
the
vertices,
edges,
and
branches
of
Γ
as
vertices,
edges,
and
branches
of
Γ;
write
V(
Γ),
E(
Γ),
B(
Γ),
respectively,
for
the
sets
of
vertices,
edges,
and
branches
of
Γ.
Also,
whenever
Γ
satisfies
a
property
of
graphs
[such
as
“finiteness”],
we
shall
say
that
Γ
satisfies
this
property.
We
shall
refer
to
the
oriented
graph
Γ
opp
obtained
from
Γ
by
reversing
the
ordering
on
the
branches
of
each
edge
as
the
opposite
oriented
graph
to
Γ.
A
morphism
of
oriented
graphs
is
defined
to
be
a
morphism
of
the
underlying
graphs
[cf.,
e.g.,
[Mzk13],
§1,
the
discussion
at
the
beginning
of
§1]
that
is
compatible
with
the
orderings
on
the
26
SHINICHI
MOCHIZUKI
edges.
Note
that
any
countably
ordered
set
E
may
be
regarded
as
an
oriented
graph
—
i.e.,
whose
vertices
are
the
elements
of
E,
whose
edges
are
the
pairs
of
adjacent
elements
of
E,
and
whose
branches
are
equipped
with
the
[total]
ordering
induced
by
the
ordering
of
E.
We
shall
refer
to
an
oriented
graph
that
arises
from
a
countably
ordered
set
as
linear.
We
shall
refer
to
the
vertex
of
a
linear
oriented
graph
Γ
determined
by
a
minimal
(respectively,
maximal)
element
of
the
corresponding
countably
ordered
set
as
the
minimal
vertex
(respectively,
maximal
vertex)
of
Γ.
Let
Γ
be
an
oriented
graph.
Then
we
shall
refer
to
as
a
pre-path
[of
length
n]
[where
n
≥
0
is
an
integer]
on
Γ
a
morphism
γ
:
Γ
γ
→
Γ,
where
Γ
γ
is
a
finite
linear
oriented
graph
with
precisely
n
edges;
we
shall
refer
to
as
a
path
[of
length
n]
on
Γ
any
isomorphism
class
[γ]
in
the
category
of
oriented
graphs
over
Γ
of
a
pre-path
γ
[of
length
n].
Write
Ω(
Γ)
for
the
set
[i.e.,
since
we
are
working
with
isomorphism
classes!]
of
paths
on
Γ.
If
γ
:
Γ
γ
→
Γ
is
a
pre-path
on
Γ,
then
we
shall
refer
to
the
image
of
the
minimal
(respectively,
maximal)
vertex
of
Γ
γ
as
the
initial
(respectively,
terminal)
vertex
of
γ,
[γ].
Two
[pre-]paths
with
the
same
initial
(respectively,
terminal;
initial
and
terminal)
vertices
will
be
referred
to
as
co-initial
(respectively,
co-terminal;
co-
verticial).
If
γ
1
,
γ
2
are
pre-paths
on
Γ
such
the
initial
vertex
of
γ
2
is
equal
to
the
terminal
vertex
of
γ
1
,
then
one
may
form
the
composite
pre-path
γ
2
◦
γ
1
[in
the
def
evident
sense],
as
well
as
the
composite
path
[γ
2
]
◦
[γ
1
]
=
[γ
2
◦
γ
1
].
Thus,
the
length
of
γ
2
◦
γ
1
is
equal
to
the
sum
of
the
lengths
of
γ
1
,
γ
2
.
Next,
let
E
⊆
Ω(
Γ)
×
Ω(
Γ)
be
a
set
of
ordered
pairs
of
paths
on
an
oriented
graph
Γ.
Then
we
shall
say
that
E
is
saturated
if
the
following
conditions
are
satisfied:
(a)
(Partial
Inclusion
of
the
Diagonal)
If
([γ
1
],
[γ
2
])
∈
E,
then
E
contains
([γ
1
],
[γ
1
])
and
([γ
2
],
[γ
2
]).
(b)
(Co-verticiality)
If
([γ
1
],
[γ
2
])
∈
E,
then
[γ
1
],
[γ
2
]
are
co-verticial.
(c)
(Transitivity)
If
([γ
1
],
[γ
2
])
∈
E
and
([γ
2
],
[γ
3
])
∈
E,
then
([γ
1
],
[γ
3
])
∈
E.
(d)
(Pre-composition)
If
([γ
1
],
[γ
2
])
∈
E
and
[γ
3
]
∈
Ω(
Γ),
then
([γ
1
]◦[γ
3
],
[γ
2
]◦
[γ
3
])
∈
E,
whenever
these
composite
paths
are
defined.
(e)
(Post-composition)
If
([γ
1
],
[γ
2
])
∈
E
and
[γ
3
]
∈
Ω(
Γ),
then
([γ
3
]
◦
[γ
1
],
[γ
3
]
◦
[γ
2
])
∈
E,
whenever
these
composite
paths
are
defined.
We
shall
say
that
E
is
symmetrically
saturated
if
E
is
saturated
and,
moreover,
satisfies
the
following
condition:
(f)
(Reflexivity)
If
([γ
1
],
[γ
2
])
∈
E,
then
([γ
2
],
[γ
1
])
∈
E.
Thus,
the
set
of
all
co-verticial
pairs
of
paths
Covert(
Γ)
⊆
Ω(
Γ)
×
Ω(
Γ)
TOPICS
IN
ABSOLUTE
ANABELIAN
GEOMETRY
III
27
is
symmetrically
saturated.
Moreover,
the
property
of
being
saturated
(respectively,
symmetrically
saturated)
is
closed
with
respect
to
forming
arbitrary
intersections
of
subsets
of
Ω(
Γ)
×
Ω(
Γ).
In
particular,
given
any
subset
E
⊆
Covert(
Γ),
it
makes
sense
to
speak
of
the
saturation
(respectively,
symmetric
saturation)
of
E
—
i.e.,
the
smallest
saturated
(respectively,
symmetrically
saturated)
subset
of
Covert(
Γ)
containing
E.
Let
Γ
be
an
oriented
graph.
Then
we
shall
refer
to
a
vertex
v
of
Γ
as
a
nexus
of
Γ
if
the
following
conditions
are
satisfied:
(a)
the
oriented
graph
Γ
v
obtained
by
removing
from
Γ
the
vertex
v,
together
with
all
of
the
edges
that
abut
to
v,
decomposes
as
a
disjoint
union
of
two
nonempty
oriented
graphs
Γ
<v
,
Γ
>v
;
(b)
every
edge
of
Γ
that
is
not
contained
in
Γ
v
either
runs
from
a
vertex
of
Γ
<v
to
v
or
from
v
to
a
vertex
of
Γ
>v
.
In
this
situation,
we
shall
refer
to
the
oriented
subgraph
Γ
≤v
(respectively,
Γ
≥v
)
consisting
of
v,
Γ
<v
(respectively,
Γ
>v
),
and
all
of
the
edges
of
Γ
that
run
to
(respectively,
emanate
from)
v
as
the
pre-nexus
portion
(respectively,
post-nexus
portion)
of
Γ.
Categories:
Let
C,
C
be
categories.
Then
we
shall
use
the
notation
Ob(C);
Arr(C)
to
denote,
respectively,
the
objects
and
arrows
of
C.
We
shall
refer
to
a
functor
φ
:
C
→
C
as
rigid
if
every
automorphism
of
φ
is
equal
to
the
identity
[cf.
[Mzk16],
§0].
If
the
identity
functor
of
C
is
rigid,
then
we
shall
say
that
C
is
id-rigid.
Let
C
be
a
category
and
Γ
an
oriented
graph.
Then
we
shall
refer
to
as
a
Γ-diagram
{A
v
,
φ
e
}
in
C
a
collection
of
data
as
follows:
(a)
for
each
v
∈
V(
Γ),
an
object
A
v
of
C;
(b)
for
each
e
∈
E(
Γ)
that
runs
from
v
1
∈
V(
Γ)
to
v
2
∈
V(
Γ),
a
morphism
φ
e
:
A
v
1
→
A
v
2
of
C.
A
morphism
{A
v
,
φ
e
}
→
{A
v
,
φ
e
}
of
Γ-diagrams
in
C
is
defined
to
be
a
collection
of
morphisms
ψ
v
:
A
v
→
A
v
for
each
vertex
v
of
Γ
that
are
compatible
with
the
φ
e
,
φ
e
.
We
shall
refer
to
an
Γ-diagram
in
C
as
commutative
if
the
composite
morphisms
determined
by
any
co-verticial
pair
of
paths
on
Γ
coincide.
Write
C[
Γ]
for
the
category
of
commutative
Γ-diagrams
in
C
and
morphisms
of
Γ-diagrams
in
C.
If
C
1
,
C
2
,
and
D
are
categories,
and
Φ
1
:
C
1
→
D;
Φ
2
:
C
2
→
D
28
SHINICHI
MOCHIZUKI
are
functors,
then
we
define
the
“categorical
fiber
product”
[cf.
[Mzk16],
§0]
C
1
×
D
C
2
of
C
1
,
C
2
over
D
to
be
the
category
whose
objects
are
triples
∼
(A
1
,
A
2
,
α
:
Φ
1
(A
1
)
→
Φ
2
(A
2
))
where
A
i
∈
Ob(C
i
)
(for
i
=
1,
2),
α
is
an
isomorphism
of
D;
and
whose
morphisms
∼
∼
(A
1
,
A
2
,
α
:
Φ
1
(A
1
)
→
Φ
2
(A
2
))
→
(B
1
,
B
2
,
β
:
Φ
1
(B
1
)
→
Φ
2
(B
2
))
are
pairs
of
morphisms
γ
i
:
A
i
→
B
i
[in
C
i
,
for
i
=
1,
2]
such
that
β
◦
Φ
1
(γ
1
)
=
Φ
2
(γ
2
)◦α.
One
verifies
easily
that
if
Φ
2
is
an
equivalence,
then
the
natural
projection
functor
C
1
×
D
C
2
→
C
1
is
also
an
equivalence.
We
shall
use
the
prefix
“ind-”
(respectively,
“pro-”)
to
mean,
strictly
speaking,
a(n)
inductive
(respectively,
projective)
system
indexed
by
an
ordered
set
isomor-
phic
to
the
positive
(respectively,
negative)
integers,
with
their
usual
ordering.
To
simplify
notation,
however,
we
shall
often
denote
“ind-objects”
via
the
correspond-
ing
“limit
objects”,
when
there
is
no
fear
of
confusion.
Let
C
be
a
category.
Then
we
shall
refer
to
a
pair
(S,
A),
where
S
∈
Ob(C),
and
A
⊆
Aut
C
(S)
is
a
subgroup,
as
a
pre-orbi-object
of
C.
[Thus,
we
think
of
the
pair
(S,
A)
as
representing
the
“stack-theoretic
quotient
of
S
by
A”.]
A
morphism
of
pre-orbi-objects
(S
1
,
A
1
)
→
(S
2
,
A
2
)
is
an
A
2
-orbit
of
morphisms
S
1
→
S
2
[relative
to
the
action
of
A
2
on
the
codomain]
that
is
closed
under
the
action
of
A
1
[on
the
domain].
We
shall
refer
to
as
an
orbi-object
{(S
ι
,
A
ι
);
α
ι,ι
}
ι,ι
∈I
any
collection
of
data
consisting
of
pre-orbi-objects
(S
ι
,
A
ι
),
which
we
shall
refer
to
as
representatives
[of
the
given
orbi-object],
together
with
“gluing
isomorphisms”
∼
α
ι,ι
:
(S
ι
,
A
ι
)
→
(S
ι
,
A
ι
)
of
pre-orbi-objects
satisfying
the
cocycle
conditions
α
ι,ι
=
α
ι
,ι
◦
α
ι,ι
,
for
ι,
ι
,
ι
∈
I.
A
morphism
of
orbi-objects
is
defined
to
be
a
collection
of
morphisms
of
pre-orbi-objects
from
each
representative
of
the
domain
to
each
representative
of
the
codomain
which
are
compatible
with
the
gluing
isomorphisms.
The
category
of
orbi-objects
associated
to
C
is
the
category
—
which
we
shall
denote
Orb(C)
—
whose
objects
are
the
orbi-objects
of
C,
and
whose
morphisms
are
the
morphisms
of
orbi-objects.
Thus,
an
object
may
be
regarded
as
a
pre-orbi-object
whose
group
of
automorphisms
is
trivial;
a
pre-orbi-object
may
be
regarded
as
an
orbi-object
with
precisely
one
representative.
In
particular,
we
obtain
a
natural
functor
C
→
Orb(C)
which
is
“functorial”
[in
the
evident
sense]
with
respect
to
C.
TOPICS
IN
ABSOLUTE
ANABELIAN
GEOMETRY
III
29
Section
1:
Galois-theoretic
Reconstruction
Algorithms
In
the
present
§1,
we
apply
the
technique
of
Belyi
cuspidalization
developed
in
[Mzk21],
§3,
to
give
a
group-theoretic
reconstruction
algorithm
[cf.
Theorem
1.9,
Corollary
1.10]
for
hyperbolic
orbicurves
of
strictly
Belyi
type
[cf.
[Mzk21],
Definition
3.5]
over
sub-p-adic
fields
that
is
compatible
with
base-change
of
the
base
field.
In
the
case
of
function
fields,
this
reconstruction
algorithm
reduces
to
a
much
more
elementary
algorithm
[cf.
Theorem
1.11],
which
is
valid
over
somewhat
more
general
base
fields,
namely
base
fields
which
are
“Kummer-faithful”
[cf.
Definition
1.5].
Let
X
be
a
hyperbolic
curve
over
a
field
k.
Write
K
X
for
the
function
field
of
X.
Then
the
content
of
following
result
is
a
consequence
of
the
well-known
theory
of
divisors
on
algebraic
curves.
Proposition
1.1.
(Review
of
Linear
Systems)
Suppose
that
X
is
proper,
and
that
k
is
algebraically
closed.
Write
Div(f
)
for
the
divisor
[of
zeroes
minus
poles
on
X]
of
f
∈
K
X
.
If
E
is
a
divisor
on
X,
then
let
us
write
Γ
×
(E)
=
{f
∈
K
X
|
Div(f
)
+
E
≥
0}
def
[where
we
use
the
notation
“(−)
≥
0”
to
denote
the
effectivity
of
the
divisor
“(−)”],
def
l(E)
=
dim
k
(Γ(X,
O
X
(E))).
Let
D
be
a
divisor
on
X.
Then:
(i)
Γ
×
(D)
admits
a
natural
free
action
by
k
×
whenever
it
is
nonempty;
there
∼
is
a
natural
bijection
Γ
×
(D)
→
Γ(X,
O
X
(D))\{0}
that
is
compatible
with
the
k
×
-
actions
on
either
side,
whenever
the
sets
of
the
bijection
are
nonempty.
(ii)
The
integer
l(D)
≥
0
is
equal
to
the
smallest
nonnegative
integer
d
such
that
there
exists
an
effective
divisor
E
of
degree
d
on
X
for
which
Γ
×
(D
−
E)
=
∅.
In
particular,
l(D)
=
0
if
and
only
if
Γ
×
(D)
=
∅.
Proposition
1.2.
(Additive
Structure
via
Linear
Systems)
Let
X,
k
be
as
in
Proposition
1.1.
Then:
(i)
There
exist
distinct
points
x,
y
1
,
y
2
∈
X(k),
together
with
a
divisor
D
on
X
such
that
x,
y
1
,
y
2
∈
Supp(D)
[where
we
write
Supp(D)
for
the
support
of
D],
such
that
l(D)
=
2,
l(D
−
E)
=
0,
for
any
effective
divisor
E
=
e
1
+
e
2
,
where
e
1
=
e
2
,
{e
1
,
e
2
}
⊆
{x,
y
1
,
y
2
}.
(ii)
Let
x,
y
1
,
y
2
,
D
be
as
in
(i).
Then
for
i
=
1,
2,
λ
∈
k
×
,
there
ex-
ists
a
unique
element
f
λ,i
∈
Γ
×
(D)
⊆
K
X
such
that
f
λ,i
(x)
=
λ,
f
λ,i
(y
i
)
=
0,
f
λ,i
(y
3−i
)
=
0.
(iii)
Let
x,
y
1
,
y
2
,
D
be
as
in
(i);
λ,
μ
∈
k
×
such
that
λ/μ
=
−1;
f
λ,1
∈
Γ
×
(D)
⊆
K
X
,
f
μ,2
∈
Γ
×
(D)
⊆
K
X
as
in
(ii).
Then
f
λ,1
+
f
μ,2
∈
Γ
×
(D)
⊆
K
X
30
SHINICHI
MOCHIZUKI
may
be
characterized
as
the
unique
element
g
∈
Γ
×
(D)
⊆
K
X
such
that
g(y
1
)
=
f
λ,1
(y
1
),
g(y
2
)
=
f
μ,2
(y
2
).
In
particular,
in
this
situation,
the
element
λ
+
μ
∈
k
×
may
be
characterized
as
the
element
g(x)
∈
k
×
.
Proof.
First,
we
consider
assertion
(i).
Let
D
be
any
divisor
on
X
such
that
l(D)
≥
2.
By
subtracting
an
appropriate
effective
divisor
from
D,
we
may
assume
that
l(D)
=
2.
Then
take
x
∈
X(k)\Supp(D)
to
be
any
point
such
that
O
X
(D)
admits
a
global
section
that
does
not
vanish
at
x
[so
l(D
−
x)
=
1];
take
y
1
∈
X(k)\(Supp(D)
{x})
to
be
any
point
such
that
O
X
(D−x)
admits
a
global
section
that
does
not
vanish
at
y
1
[so
l(D
−
x
−
y
1
)
=
0,
which
implies
that
l(D
−
y
1
)
=
1];
take
y
2
∈
X(k)\(Supp(D)
{x,
y
1
})
to
be
any
point
such
that
O
X
(D
−
x),
O
X
(D
−
y
1
)
admit
global
sections
that
do
not
vanish
at
y
2
[so
l(D
−
x
−
y
2
)
=
l(D
−
y
1
−
y
2
)
=
0].
This
completes
the
proof
of
assertion
(i).
Now
assertions
(ii),
(iii)
follow
immediately
from
assertion
(i).
The
following
reconstruction
of
the
additive
structure
from
divisors
and
rational
functions
is
implicit
in
the
argument
of
[Uchi],
§3,
Lemmas
8-11
[cf.
also
[Tama],
Lemma
4.7].
Proposition
1.3.
(Additive
Structure
via
Valuation
and
Evaluation
Maps)
Let
X,
k
be
as
in
Proposition
1.1.
Then
there
exists
a
functorial
algo-
×
{0}
[i.e.,
arising
from
rithm
for
constructing
the
additive
structure
on
K
X
the
field
structure
of
K
X
]
from
the
following
data:
×
;
(a)
the
[abstract!]
group
K
X
(b)
the
set
of
[surjective]
homomorphisms
×
Z}
x∈X(k)
V
X
=
{ord
x
:
K
X
def
∼
[so
we
have
a
natural
bijection
V
X
→
X(k)]
that
arise
as
valuation
maps
associated
to
points
x
∈
X(k);
×
(c)
for
each
homomorphism
v
=
ord
x
∈
V
X
,
the
subgroup
U
v
⊆
K
X
given
×
by
the
f
∈
K
X
such
that
f
(x)
=
1.
Here,
the
term
“functorial”
is
with
respect
to
isomorphisms
[in
the
evident
sense]
of
such
triples
[i.e.,
consisting
of
a
group,
a
set
of
homomorphisms
from
the
group
to
Z,
and
a
collection
of
subgroups
of
the
group
parametrized
by
elements
of
this
set
of
homomorphisms]
arising
from
proper
hyperbolic
curves
[i.e.,
“X”]
over
alge-
braically
closed
fields
[i.e.,
“k”].
×
may
be
constructed
as
the
inter-
Proof.
Indeed,
first
we
observe
that
k
×
⊆
K
X
section
v∈V
X
Ker(v).
Since,
for
v
∈
V
X
,
we
have
a
direct
product
decomposition
Ker(v)
=
U
v
×
k
×
,
the
projection
to
k
×
allows
us
to
“evaluate”
elements
of
Ker(v)
[i.e.,
“functions
that
are
invertible
at
the
point
associated
to
v”],
so
as
to
obtain
“values”
of
such
elements
∈
k
×
.
Next,
let
us
observe
that
the
set
of
homomorphisms
TOPICS
IN
ABSOLUTE
ANABELIAN
GEOMETRY
III
31
V
X
of
(b)
allows
one
to
speak
of
divisors
and
effective
divisors
associated
to
[the
×
×
abstract
group]
K
X
of
(a).
If
D
is
a
divisor
associated
to
K
X
,
then
we
may
define
×
Γ
(D)
as
in
Proposition
1.1,
and
hence
compute
the
integer
l(D)
as
in
Proposition
1.1,
(ii).
In
particular,
it
makes
sense
to
speak
of
data
as
in
Proposition
1.2,
(i),
associated
to
the
abstract
data
(a),
(b),
(c).
Thus,
by
evaluating
elements
of
vari-
ous
Ker(v),
for
v
∈
V
X
,
we
may
apply
the
characterizations
of
Proposition
1.2,
(ii),
(iii),
to
construct
the
additive
structure
of
k
×
,
hence
also
the
additive
structure
of
×
K
X
[i.e.,
by
“evaluating”
at
various
v
∈
V
X
].
Remark
1.3.1.
Note
that
if
G
is
an
abstract
group,
then
the
datum
of
a
surjection
def
v
:
G
Z
may
be
thought
of
as
the
datum
of
a
subgroup
H
=
Ker(v),
together
∼
with
the
datum
of
a
choice
of
generator
of
the
quotient
group
G/H
→
Z.
Proposition
1.4.
(Synchronization
of
Geometric
Cyclotomes)
Suppose
that
X
is
proper,
and
that
k
is
of
characteristic
zero.
If
U
⊆
X
is
a
nonempty
open
subscheme,
then
we
have
a
natural
exact
sequence
of
profinite
groups
1
→
Δ
U
→
Π
U
→
G
k
→
1
def
def
—
where
we
write
Π
U
=
π
1
(U
)
→
G
k
=
π
1
(Spec(k))
for
the
natural
surjection
of
étale
fundamental
groups
[relative
to
some
choice
of
basepoints],
Δ
U
for
the
kernel
of
this
surjection.
Then:
def
(i)
Let
U
⊆
X
be
a
nonempty
open
subscheme,
x
∈
X(k)\U
(k),
U
x
=
X\{x}
⊆
X.
Then
the
inertia
group
I
x
of
x
in
Δ
U
is
naturally
isomorphic
to
Z(1);
the
kernels
of
the
natural
surjections
Δ
U
Δ
U
x
,
Π
U
Π
U
x
are
topolog-
ically
normally
generated
by
the
inertia
groups
of
points
of
U
x
\U
[each
of
which
is
naturally
isomorphic
to
Z(1)].
(ii)
Let
x,
U
x
be
as
in
(i).
Then
we
have
a
natural
exact
sequence
of
profinite
groups
1
→
I
x
→
Δ
c-cn
U
x
→
Δ
X
→
1
—
where
we
write
Δ
U
x
Δ
c-cn
U
x
for
the
maximal
cuspidally
central
quotient
of
Δ
U
x
[i.e.,
the
maximal
intermediate
quotient
Δ
U
x
Q
Δ
X
such
that
Ker(Q
Δ
X
)
lies
in
the
center
of
Q
—
cf.
[Mzk19],
Definition
1.1,
(i)].
Moreover,
applying
the
differential
of
the
“E
2
-term”
of
the
Leray
spectral
sequence
associated
to
this
group
extension
to
the
element
=
Hom(I
x
,
I
x
)
=
H
0
(Δ
X
,
H
1
(I
x
,
I
x
))
1
∈
Z
yields
an
element
∈
H
2
(Δ
X
,
H
0
(I
x
,
I
x
))
=
Hom(M
X
,
I
x
),
where
we
write
def
Z)
M
X
=
Hom(H
2
(Δ
X
,
Z),
[cf.
the
discussion
at
the
beginning
of
[Mzk19],
§1];
this
last
element
corresponds
to
the
natural
isomorphism
∼
M
X
→
I
x
32
SHINICHI
MOCHIZUKI
[relative
to
the
well-known
natural
identifications
of
I
x
,
M
X
with
Z(1)
—
cf.,
e.g.,
(i)
above;
[Mzk19],
Proposition
1.2,
(i)].
In
particular,
this
yields
a
“purely
group-theoretic
algorithm”
[cf.
Remark
1.9.8
below
for
more
on
the
meaning
of
this
terminology]
for
constructing
this
isomorphism
from
the
surjection
Δ
U
x
Δ
X
.
Proof.
Assertion
(i)
is
well-known
[and
easily
verified
from
the
definitions].
Asser-
tion
(ii)
follows
immediately
from
[Mzk19],
Proposition
1.6,
(iii).
Definition
1.5.
Let
k
be
a
field
of
characteristic
zero,
k
an
algebraic
closure
def
of
k,
G
k
=
Gal(k/k).
Then
we
shall
say
that
k
is
Kummer-faithful
(respectively,
torally
Kummer-faithful)
if,
for
every
finite
extension
k
H
⊆
k
of
k,
where
we
write
def
H
=
Gal(k/k
H
)
⊆
G
k
,
and
every
semi-abelian
variety
(respectively,
every
torus)
A
over
k
H
,
either
of
the
following
two
equivalent
conditions
is
satisfied:
(a)
We
have
N
·
A(k
H
)
=
{0}
N
≥1
—
where
N
ranges
over
the
positive
integers.
(b)
The
associated
Kummer
map
A(k
H
)
→
H
1
(H,
Hom(Q/Z,
A(k)))
is
an
injection.
[To
verify
the
equivalence
of
(a)
and
(b),
it
suffices
to
consider,
on
the
étale
site
of
Spec(k
H
),
the
long
exact
sequences
in
étale
cohomology
associated
to
the
exact
N
·
sequences
0
−→
N
A
−→
A
−→
A
−→
0
arising
from
multiplication
by
positive
integers
N
.]
Remark
1.5.1.
In
the
notation
of
Definition
1.5,
suppose
that
k
is
a
torally
Kummer-faithful
field,
l
a
prime
number.
Then
it
follows
immediately
from
the
injectivity
of
the
Kummer
map
associated
to
G
m
over
any
finite
extension
of
k
that
contains
a
primitive
l-th
root
of
unity
that
the
cyclotomic
character
χ
l
:
G
k
→
Z
×
l
has
open
image
[cf.
the
notion
of
“l-cyclotomic
fullness”
discussed
in
[Mzk20],
Lemma
4.5].
In
particular,
it
makes
sense
to
speak
of
the
“power-equivalence
class
of
χ
l
”
[cf.
[Mzk20],
Lemma
4.5,
(ii)]
among
characters
G
k
→
Z
×
l
—
i.e.,
the
equivalence
class
with
respect
to
the
equivalence
relation
ρ
1
∼
ρ
2
[for
characters
N
N
ρ
1
,
ρ
2
:
G
k
→
Z
×
l
]
defined
by
the
condition
that
ρ
1
=
ρ
2
for
some
positive
integer
N
.
Remark
1.5.2.
By
considering
the
Weil
restrictions
of
semi-abelian
varieties
or
tori
over
finite
extensions
of
k
to
k,
one
verifies
immediately
that
one
obtains
an
equivalent
definition
of
the
terms
“Kummer-faithful”
and
“torally
Kummer-
faithful”
if,
in
Definition
1.5,
one
restricts
k
H
to
be
equal
to
k.
Remark
1.5.3.
In
the
following
discussion,
if
k
is
a
field,
then
we
denote
the
subgroup
of
roots
of
unity
of
k
×
by
μ(k)
⊆
k
×
.
TOPICS
IN
ABSOLUTE
ANABELIAN
GEOMETRY
III
33
(i)
Let
k
be
a(n)
[not
necessarily
finite!]
algebraic
field
extension
of
a
number
field
such
that
there
exists
a
nonarchimedean
prime
of
k
that
is
unramified
over
some
number
field
contained
in
k,
and,
moreover,
for
every
finite
extension
k
†
of
k,
μ(k
†
)
is
finite.
Then
I
claim
that:
k
is
torally
Kummer-faithful.
Indeed,
since
[as
one
verifies
immediately]
any
finite
extension
of
k
satisfies
the
same
hypotheses
as
k,
one
verifies
immediately
that
it
suffices
to
show
that
N
(k
×
)
N
=
{1}
[where
N
ranges
over
the
positive
integers].
Let
f
∈
N
(k
×
)
N
.
If
f
∈
μ(k),
then
the
assumption
concerning
μ(k)
implies
immediately
that
f
=
1;
thus,
we
may
assume
without
loss
of
generality
that
f
∈
μ(k).
But
then
there
exists
a
nonarchimedean
prime
p
of
k
that
is
unramified
over
some
number
field
contained
in
k.
In
particular,
if
we
write
k
p
for
the
completion
of
k
at
p
and
p
for
the
residue
characteristic
of
p,
then
k
p
embeds
into
a
finite
extension
of
the
quotient
field
of
the
ring
of
Witt
vectors
of
an
algebraic
closure
of
F
p
.
Thus,
the
fact
that
f
admits
arbitrary
p-power
roots
in
k
p
yields
a
contradiction.
This
completes
the
proof
of
the
claim.
(ii)
It
follows
immediately
from
the
definitions
that
“Kummer-faithful
=⇒
torally
Kummer-faithful”.
On
the
other
hand,
as
was
pointed
out
to
the
author
by
A.
Tamagawa,
one
may
construct
an
example
of
a
field
which
is
torally
Kummer-
faithful,
but
not
Kummer-faithful,
as
follows:
Let
E
be
√
an
elliptic
curve
over
a
number
field
k
0
that
admits
complex
multiplication
by
Q(
−1)
and,
moreover,
has
good
reduction
at
every
nonarchimedean
prime
of
k
0
.
Let
k
be
an
algebraic
closure
def
of
k
0
,
G
k
0
=
Gal(k/k
0
),
p
a
prime
number
≡
1
(mod
4).
Write
V
for
the
p-adic
Tate
module
associated
to
E.
Thus,
the
G
k
0
-module
V
decomposes
[since
p
≡
1
(mod
4)]
into
a
direct
sum
W
⊕
W
of
submodules
W,
W
⊆
V
of
rank
one.
Write
χ
:
G
k
0
→
Z
×
p
for
the
character
determined
by
W
.
Thus,
[as
is
well-known]
χ
is
unramified
over
every
nonarchimedean
prime
of
k
0
of
residue
characteristic
l
=
p,
as
well
as
over
some
nonarchimedean
prime
of
k
0
of
residue
characteristic
p.
But
one
verifies
immediately
[for
instance,
by
considering
ramification
over
Q]
that
this
implies
that,
if
we
write
k
for
the
extension
field
of
k
0
determined
by
the
kernel
of
χ,
then
μ(k)
is
finite.
Thus,
since
any
finite
extension
of
k
0
satisfies
the
same
hypotheses
as
k
0
,
we
conclude
that
k
satisfies
the
hypotheses
of
(i),
so
k
is
torally
Kummer-faithful.
On
the
other
hand,
[by
the
definition
of
χ,
W
,
V
!]
the
Kummer
map
on
E(k)
is
not
injective,
so
k
is
not
Kummer-faithful.
Remark
1.5.4.
(i)
Observe
that
every
sub-p-adic
field
k
[cf.
[Mzk5],
Definition
15.4,
(i)]
is
Kummer-faithful,
i.e.,
“sub-p-adic
=⇒
Kummer-faithful”.
Indeed,
to
verify
this,
one
reduces
immediately,
by
base-change,
to
the
case
where
k
is
a
finitely
generated
extension
of
an
MLF,
which
may
be
thought
of
as
the
function
field
of
a
variety
over
an
MLF.
Then
by
restricting
to
various
closed
points
of
this
variety,
one
reduces
to
the
case
where
k
itself
is
an
MLF.
On
the
other
hand,
if
k,
hence
also
k
H
[cf.
the
notation
of
Definition
1.5],
is
a
finite
extension
of
Q
p
,
then
A(k
H
)
is
an
extension
of
a
finitely
generated
Z-module
by
a
compact
abelian
p-adic
Lie
group,
hence
contains
an
open
subgroup
that
is
isomorphic
to
a
finite
product
of
copies
of
Z
p
.
In
particular,
the
condition
of
Definition
1.5,
(a),
is
satisfied.
34
SHINICHI
MOCHIZUKI
(ii)
A
similar
argument
to
the
argument
of
(i)
shows
that
every
finitely
gen-
erated
extension
of
a
Kummer-faithful
field
(respectively,
torally
Kummer-faithful
field)
is
itself
Kummer-faithful
(respectively,
torally
Kummer-faithful
field).
(iii)
On
the
other
hand,
observe
that
if,
for
instance,
I
is
an
infinite
set,
then
the
def
field
k
=
Q
p
(x
i
)
i∈I
[which
is
not
a
finitely
generated
extension
of
Q
p
]
constitutes
an
example
of
a
Kummer-faithful
field
which
is
not
sub-p-adic.
Indeed,
if,
for
H,
A
as
in
Definition
1.5,
0
=
f
∈
A(k
H
)
lies
in
the
kernel
of
the
associated
Kummer
map,
then
observe
that
there
exists
some
finite
subset
I
⊆
I
such
that
if
we
set
def
k
=
Q
p
(x
i
)
i∈I
,
then,
for
some
finite
extension
k
H
⊆
k
H
of
k
,
we
may
assume
,
that
f
∈
A
(k
H
)
⊆
A(k
H
),
that
A
descends
to
a
semi-abelian
variety
A
over
k
H
def
(x
i
)
i∈I
,
where
we
set
I
=
I\I
.
Since
k
H
is
algebraically
closed
and
that
k
H
=
k
H
in
k
H
,
it
thus
follows
that
all
roots
of
f
defined
over
k
H
are
in
fact
defined
over
.
Thus,
the
existence
of
f
contradicts
the
fact
that
the
sub-p-adic
field
k
H
is
k
H
Kummer-faithful.
Finally,
to
see
that
k
is
not
sub-p-adic,
suppose
that
k
⊆
K,
where
K
is
a
finitely
generated
extension
of
an
MLF
K
0
of
residue
characteristic
p
0
such
that
K
0
is
algebraically
closed
in
K.
Let
l
=
p,
p
0
be
a
prime
number.
Then
Q
p
⊇
k
∗
=
(k
×
)
l
⊆
K
∗
=
N
def
(K
×
)
l
⊆
K
0
N
def
l
N
l
N
—
where
one
verifies
immediately
that
the
additive
group
generated
by
k
∗
(respec-
tively,
K
∗
)
in
k
(respectively,
K)
forms
a
compact
open
neighborhood
of
0
in
Q
p
(respectively,
K
0
).
In
particular,
it
follows
that
the
inclusion
k
→
K
determines
a
continuous
homomorphism
of
topological
fields
Q
p
→
K
0
.
But
this
implies
imme-
diately
that
p
0
=
p,
and
that
Q
p
→
K
0
is
a
Q
p
-algebra
homomorphism.
Thus,
the
theory
of
transcendence
degree
yields
a
contradiction
[for
instance,
by
considering
the
morphism
on
Kähler
differentials
induced
by
k
→
K].
(iv)
One
verifies
immediately
that
the
generalized
sub-p-adic
fields
of
[Mzk8],
Definition
4.11,
are
not,
in
general,
torally
Kummer-faithful.
Proposition
1.6.
(Kummer
Classes
of
Rational
Functions)
In
the
situ-
ation
of
Proposition
1.4,
suppose
further
that
k
is
a
Kummer-faithful
field.
If
U
⊆
X
is
a
nonempty
open
subscheme,
then
let
us
write
×
)
→
H
1
(Π
U
,
M
X
)
κ
U
:
Γ(U,
O
U
—
where
M
X
∼
is
as
in
Proposition
1.4,
(ii)
—
for
the
associated
Kummer
=
Z(1)
map
[cf.,
e.g.,
the
discussion
at
the
beginning
of
[Mzk19],
§2].
Also,
for
d
∈
Z,
let
us
write
J
d
→
Spec(k)
for
the
connected
component
of
the
Picard
scheme
of
X
→
Spec(k)
that
parametrizes
line
bundles
of
degree
d
[cf.,
e.g.,
the
discussion
def
def
preceding
[Mzk19],
Proposition
2.2];
J
=
J
0
;
Π
J
d
=
π
1
(J
d
).
[Thus,
we
have
a
natural
morphism
X
→
J
1
that
sends
a
point
of
X
to
the
line
bundle
of
degree
1
associated
to
the
point;
this
morphism
induces
a
surjection
Π
X
Π
J
1
on
étale
fundamental
groups
whose
kernel
is
equal
to
the
commutator
subgroup
of
Δ
X
.]
Then:
(i)
The
Kummer
map
κ
U
is
injective.
TOPICS
IN
ABSOLUTE
ANABELIAN
GEOMETRY
III
35
(ii)
For
x
∈
X(k),
write
s
x
:
G
k
→
Π
X
for
the
associated
section
[well-
defined
up
to
conjugation
by
Δ
X
],
t
x
:
G
k
→
Π
J
1
for
the
composite
of
s
x
with
the
natural
surjection
Π
X
Π
J
1
.
Then
for
any
divisor
D
of
degree
d
on
X
such
that
Supp(D)
⊆
X(k),
forming
the
appropriate
Z-linear
combination
of
“t
x
’s”
for
x
∈
Supp(D)
[cf.,
e.g.,
the
discussion
preceding
[Mzk19],
Proposition
2.2]
yields
a
section
t
D
:
G
k
→
Π
J
d
;
if,
moreover,
d
=
0,
then
t
D
:
G
k
→
Π
J
coincides
[up
to
conjugation
by
Δ
X
]
with
the
section
determined
by
the
identity
element
∈
J(k)
if
and
only
if
the
divisor
D
is
principal.
(iii)
Suppose
that
U
=
X\S,
where
S
⊆
X(k)
is
a
finite
subset.
Then
restrict-
ing
cohomology
classes
of
Π
U
to
the
various
I
x
[cf.
Proposition
1.4,
(i)],
for
x
∈
S,
yields
a
natural
exact
sequence
×
∧
1
1
→
(k
)
→
H
(Π
U
,
M
X
)
→
Z
x∈S
∼
via
the
isomorphism
I
x
→
(I
,
M
X
)
with
Z
M
X
of
—
where
we
identify
Hom
Z
x
Proposition
1.4,
(ii);
(k
×
)
∧
denotes
the
profinite
completion
of
k
×
.
Moreover,
the
×
image
[via
κ
U
]
of
Γ(U,
O
U
)
in
H
1
(Π
U
,
M
X
)/(k
×
)
∧
is
equal
to
the
inverse
image
in
H
1
(Π
U
,
M
X
)/(k
×
)
∧
of
the
submodule
of
Z
Z
⊆
x∈S
x∈S
determined
by
the
principal
divisors
[with
support
in
S].
Proof.
Assertion
(i)
follows
immediately
[by
restricting
to
smaller
and
smaller
“U
’s”]
from
the
fact
[cf.
Remark
1.5.4,
(ii)]
that
since
k
is
[torally]
Kummer-
faithful,
so
is
the
function
field
K
X
of
X.
Assertion
(ii)
follows
from
the
argument
of
[Mzk19],
Proposition
2.2,
(i),
together
with
the
assumption
that
k
is
Kummer-
faithful.
As
for
assertion
(iii),
just
as
in
the
proof
of
[Mzk19],
Proposition
2.1,
(ii),
to
verify
assertion
(iii),
it
suffices
to
verify
that
H
0
(G
k
,
Δ
ab
X
)
=
0;
but,
in
light
of
and
the
torsion
points
of
the
Jacobian
J,
the
well-known
relationship
between
Δ
ab
X
0
ab
the
fact
that
H
(G
k
,
Δ
X
)
=
0
follows
immediately
from
our
assumption
that
k
is
Kummer-faithful
[cf.
the
argument
applied
to
G
m
in
Remark
1.5.1].
Definition
1.7.
Suppose
that
k
is
of
characteristic
zero.
Let
k
be
an
algebraic
closure
of
k;
write
k
NF
⊆
k
for
the
[“number
field”]
algebraic
closure
of
Q
in
k.
def
(i)
We
shall
say
that
X
is
an
NF-curve
if
X
k
=
X
×
k
k
is
defined
over
k
NF
[cf.
Remark
1.7.1
below].
(ii)
Suppose
that
X
is
an
NF-curve.
Then
we
shall
refer
to
points
of
X(k)
(respectively,
rational
functions
on
X
k
;
constant
rational
functions
on
X
k
[i.e.,
which
arise
from
elements
of
k])
that
descend
to
k
NF
[cf.
Remark
1.7.1
below]
as
NF-points
of
(respectively,
NF-rational
functions
on;
NF-constants
on)
X
k
.
Remark
1.7.1.
Suppose
that
X
is
of
type
(g,
r).
Then
observe
that
X
is
an
NF-curve
if
and
only
if
the
k-valued
point
of
the
moduli
stack
of
hyperbolic
curves
36
SHINICHI
MOCHIZUKI
of
type
(g,
r)
over
Q
determined
by
X
arises,
in
fact,
from
a
k
NF
-valued
point.
In
particular,
one
verifies
immediately
that
if
X
is
an
NF-curve,
then
the
descent
data
of
X
k
from
k
to
k
NF
is
unique.
Proposition
1.8.
(Characterization
of
NF-Constants
and
NF-Rational
Functions)
In
the
situation
of
Proposition
1.6,
(iii),
suppose
further
that
U
[hence
also
X]
is
an
NF-curve.
Write
P
U
⊆
H
1
(Π
U
,
M
X
)
for
the
inverse
image
of
the
submodule
of
Z
Z
⊆
x∈S
x∈S
determined
by
the
cuspidal
principal
divisors
[i.e.,
principal
divisors
supported
on
the
cusps]
—
cf.
Proposition
1.6,
(iii).
Then:
(i)
A
class
η
∈
P
U
is
the
Kummer
class
of
a
nonconstant
NF-rational
function
if
and
only
if
there
exist
a
positive
multiple
η
†
of
η
and
NF-points
x
i
∈
U
(k
x
),
where
i
=
1,
2,
and
k
x
is
a
finite
extension
of
k,
such
that
the
coho-
mology
classes
def
η
†
|
x
i
=
s
∗
x
i
(η
†
)
∈
H
1
(G
k
x
,
M
X
)
—
where
we
write
s
x
i
:
G
k
x
→
Π
U
for
the
[outer]
homomorphism
determined
by
x
i
[cf.
the
notation
of
Proposition
1.6,
(ii)]
—
satisfy
η
†
|
x
1
=
0
[i.e.,
=
1,
if
one
works
multiplicatively],
η
†
|
x
2
=
0.
×
).
(ii)
Suppose
that
there
exist
nonconstant
NF-rational
functions
∈
Γ(U,
O
U
1
×
∧
∼
Then
a
class
η
∈
P
U
H
(G
k
,
M
X
)
=
(k
)
[cf.
the
exact
sequence
of
Proposition
1.6,
(iii)]
is
the
Kummer
class
of
an
NF-constant
∈
k
×
if
and
only
if
there
exist
a
×
nonconstant
NF-rational
function
f
∈
Γ(U,
O
U
)
and
an
NF-point
x
∈
U
(k
x
),
where
k
x
is
a
finite
extension
of
k,
such
that
κ
U
(f
)|
x
=
η|
G
kx
∈
H
1
(G
k
x
,
M
X
)
—
where
we
use
the
notation
“|
x
”
as
in
(i).
Proof.
Suppose
that
X
k
descends
to
a
hyperbolic
curve
X
NF
over
k
NF
.
Then
[since
k
NF
is
algebraically
closed]
any
nonconstant
rational
function
on
X
NF
determines
a
morphism
X
NF
→
P
1
k
such
that
the
induced
map
X
NF
(k
NF
)
→
P
1
k
(k
NF
)
is
NF
NF
surjective.
In
light
of
this
fact
[cf.
also
the
fact
that
U
is
also
assumed
to
be
an
NF-curve],
assertions
(i),
(ii)
follow
immediately
from
the
definitions.
Now,
by
combining
the
“reconstruction
algorithms”
given
in
the
various
results
discussed
above,
we
obtain
the
main
result
of
the
present
§1.
Theorem
1.9.
(The
NF-portion
of
the
Function
Field
via
Belyi
Cus-
pidalization
over
Sub-p-adic
Fields)
Let
X
be
a
hyperbolic
orbicurve
of
TOPICS
IN
ABSOLUTE
ANABELIAN
GEOMETRY
III
37
strictly
Belyi
type
[cf.
[Mzk21],
Definition
3.5]
over
a
sub-p-adic
field
[cf.
[Mzk5],
Definition
15.4,
(i)]
k,
for
some
prime
p;
k
an
algebraic
closure
of
k;
k
NF
⊆
k
the
algebraic
closure
of
Q
in
k;
1
→
Δ
X
→
Π
X
→
G
k
→
1
def
def
—
where
Π
X
=
π
1
(X)
→
G
k
=
Gal(k/k)
denotes
the
natural
surjection
of
étale
fundamental
groups
[relative
to
some
choice
of
basepoints],
and
Δ
X
denotes
the
kernel
of
this
surjection
—
the
resulting
extension
of
profinite
groups.
Then
there
exists
a
functorial
“group-theoretic”
algorithm
[cf.
Remark
1.9.8
below
for
more
on
the
meaning
of
this
terminology]
for
reconstructing
the
“NF-portion
of
the
function
field”
of
X
from
the
extension
of
profinite
groups
1
→
Δ
X
→
Π
X
→
G
k
→
1;
this
algorithm
consists
of
the
following
steps:
(a)
One
constructs
the
various
surjections
Π
U
Π
Y
—
where
Y
is
a
hyperbolic
[NF-]curve
that
arises
as
a
finite
étale
covering
of
X;
U
⊆
Y
is
an
open
subscheme
obtained
by
removing
an
arbitrary
def
def
finite
collection
of
NF-points;
Π
U
=
π
1
(U
);
Π
Y
=
π
1
(Y
)
⊆
Π
X
—
via
the
technique
of
“Belyi
cuspidalization”,
as
described
in
[Mzk21],
Corollary
3.7,
(a),
(b),
(c).
Here,
we
note
that
by
allowing
U
to
vary,
we
obtain
a
“group-theoretic”
construction
of
Π
U
equipped
with
the
collection
of
subgroups
that
arise
as
decomposition
groups
of
NF-points.
(b)
One
constructs
the
natural
isomorphisms
∼
def
I
z
→
μ
(Π
U
)
=
M
Z
Z
—
where
U
⊆
Y
→
X
is
as
in
(i),
Y
is
of
genus
≥
2,
Z
is
the
canonical
compactification
of
Y
,
the
points
of
Z\U
are
all
rational
over
the
base
field
k
Z
of
Z,
z
∈
(Z\U
)(k
Z
)
—
via
the
technique
of
Proposition
1.4,
(ii).
(c)
For
U
⊆
Y
⊆
Z
as
in
(b),
one
constructs
the
subgroup
P
U
⊆
H
1
(Π
U
,
μ
(Π
U
))
Z
determined
by
the
cuspidal
principal
divisors
via
the
isomorphisms
of
(b)
and
the
characterization
of
principal
divisors
given
in
Proposition
1.6,
(ii)
[cf.
also
the
decomposition
groups
of
(a);
Proposition
1.6,
(iii)].
(d)
For
U
⊆
Y
⊆
Z,
k
Z
as
in
(b),
one
constructs
the
subgroups
×
1
k
NF
⊆
K
Z
×
NF
→
lim
(Π
U
))
−→
H
(Π
V
,
μ
Z
V
—
where
V
ranges
over
the
open
subschemes
obtained
by
removing
finite
collections
of
NF-points
from
Z
×
k
Z
k
,
for
k
a
finite
extension
of
k
Z
;
def
Π
V
=
π
1
(V
);
K
Z
NF
is
the
function
field
of
the
curve
Z
NF
obtained
by
38
SHINICHI
MOCHIZUKI
descending
Z
×
k
Z
k
to
k
NF
;
the
“→”
arises
from
the
Kummer
map
—
via
the
subgroups
of
(c)
and
the
characterizations
of
Kummer
classes
of
nonconstant
NF-rational
functions
and
NF-constants
given
in
Proposition
1.8,
(i),
(ii)
[cf.
also
the
decomposition
groups
of
(a)].
(e)
One
constructs
the
additive
structure
on
×
k
NF
{0};
K
Z
×
NF
{0}
[notation
as
in
(d)]
by
applying
the
functorial
algorithm
of
Proposition
1.3
to
the
data
of
the
form
described
in
Proposition
1.3,
(a),
(b),
(c),
arising
from
the
construction
of
(d)
[cf.
also
the
decomposition
groups
of
(a),
the
isomorphisms
of
(b)].
Finally,
the
asserted
“functoriality”
is
with
respect
to
arbitrary
open
injective
homomorphisms
of
extensions
of
profinite
groups
[cf.
also
Remark
1.10.1
below],
as
well
as
with
respect
to
homomorphisms
of
extensions
of
profinite
groups
arising
from
a
base-change
of
the
base
field
[i.e.,
k].
Proof.
The
validity
of
the
algorithm
asserted
in
Theorem
1.9
is
immediate
from
the
various
results
cited
in
the
statement
of
this
algorithm.
Remark
1.9.1.
When
k
is
an
MLF
[cf.
[Mzk20],
§0],
one
verifies
immediately
that
one
may
give
a
tempered
version
of
Theorem
1.9
[cf.
[Mzk21],
Remark
3.7.1],
in
which
the
profinite
étale
fundamental
group
Π
X
is
replaced
by
the
tempered
funda-
mental
group
of
X
[and
the
expression
“profinite
group”
is
replaced
by
“topological
group”].
Remark
1.9.2.
When
k
is
an
MLF
or
NF
[cf.
[Mzk20],
§0],
the
“extension
of
profinite
groups
1
→
Δ
X
→
Π
X
→
G
k
→
1”
that
appears
in
the
input
data
for
the
algorithm
of
Theorem
1.9
may
be
replaced
by
the
single
profinite
group
Π
X
[cf.
[Mzk20],
Theorem
2.6,
(v),
(vi)].
A
similar
remark
applies
in
the
tempered
case
discussed
in
Remark
1.9.1.
Remark
1.9.3.
Note
that
unlike
the
case
with
k
NF
,
K
Z
NF
,
the
algorithm
of
Theorem
1.9
does
not
furnish
a
means
for
reconstructing
k,
K
Z
in
general
—
cf.
Corollary
1.10
below
concerning
the
case
when
k
is
an
MLF.
Remark
1.9.4.
Suppose
that
k
is
an
MLF.
Then
G
k
,
which
is
of
cohomological
dimension
2
[cf.
[NSW],
Theorem
7.1.8,
(i)],
may
be
thought
of
as
having
one
rigid
dimension
and
one
non-rigid
dimension.
Indeed,
the
maximal
unramified
quotient
∼
G
k
G
unr
=
Z
k
is
generated
by
the
Frobenius
element,
which
may
be
characterized
by
an
entirely
group-theoretic
algorithm
[hence
is
preserved
by
isomorphisms
of
absolute
Galois
groups
of
MLF’s
—
cf.
[Mzk9],
Proposition
1.2.1,
(iv)];
thus,
this
quotient
G
k
TOPICS
IN
ABSOLUTE
ANABELIAN
GEOMETRY
III
39
∼
may
be
thought
of
as
a
“rigid
dimension”.
On
the
other
hand,
the
G
unr
=
Z
k
dimension
of
G
k
represented
by
the
inertia
group
in
I
k
⊆
G
k
[which,
as
is
well-known,
is
of
cohomological
dimension
1]
is
“far
from
rigid”
—
a
phenomenon
that
may
be
seen,
for
instance,
in
the
existence
[cf.,
e.g.,
[NSW],
the
Closing
Remark
preceding
Theorem
12.2.7]
of
isomorphisms
of
absolute
Galois
groups
of
MLF’s
which
fail
[equivalently
—
cf.
[Mzk20],
Corollary
3.7]
to
be
“RF-
preserving”,
“uniformly
toral”,
or
“geometric”.
By
contrast,
it
is
interesting
to
observe
that:
The
“group-theoretic”
algorithm
of
Theorem
1.9
shows
that
the
condition
of
being
“coupled
with
Δ
X
”
[i.e.,
via
the
extension
determined
by
Π
X
]
has
the
effect
of
rigidifying
both
of
the
2
dimensions
of
G
k
[cf.
also
Corollary
1.10
below].
This
point
of
view
will
be
of
use
in
our
development
of
the
archimedean
theory
in
§2
below
[cf.,
e.g.,
Remark
2.7.3
below].
Remark
1.9.5.
(i)
Note
that
the
functoriality
with
respect
to
isomorphisms
of
the
algorithm
of
Theorem
1.9
may
be
regarded
as
yielding
a
new
proof
of
the
“profinite
absolute
version
of
the
Grothendieck
Conjecture
over
number
fields”
[cf.,
e.g.,
[Mzk15],
The-
orem
3.4]
that
does
not
logically
depend
on
the
theorem
of
Neukirch-Uchida
[cf.,
e.g.,
[Mzk15],
Theorem
3.1].
Moreover,
to
the
author’s
knowledge:
The
technique
of
Theorem
1.9
yields
the
first
logically
independent
proof
of
a
consequence
of
the
theorem
of
Neukirch-Uchida
that
involves
an
explicit
construction
of
the
number
fields
involved.
Put
another
way,
the
algorithm
of
Theorem
1.9
yields
a
proof
of
a
consequence
of
the
theorem
of
Neukirch-Uchida
on
number
fields
in
the
style
of
Uchida’s
work
on
function
fields
in
positive
characteristic
[i.e.,
[Uchi]]
—
cf.,
especially,
Proposition
1.3.
(ii)
One
aspect
of
the
theorem
of
Neukirch-Uchida
is
that
its
proof
relies
es-
sentially
on
the
data
arising
from
the
decomposition
of
primes
in
finite
extensions
of
a
number
field
—
i.e.,
in
other
words,
on
the
“global
address”
of
a
prime
among
all
the
primes
of
a
number
field.
Such
a
“global
address”
is
manifestly
annihilated
by
the
operation
of
localization
at
the
prime
under
consideration.
In
particular,
the
crucial
functoriality
of
Theorem
1.9
with
respect
to
change
of
base
field
[e.g.,
from
a
number
field
to
a
nonarchimedean
completion
of
the
number
field]
is
an-
other
reflection
of
the
way
in
which
the
nature
of
the
proof
of
Theorem
1.9
over
number
fields
differs
quite
fundamentally
from
the
essentially
global
proof
of
the
theorem
of
Neukirch-Uchida
[cf.
also
Remark
3.7.6,
(iii),
(v),
below].
This
“crucial
functoriality”
may
also
be
thought
of
as
a
sort
of
essential
independence
of
the
algorithms
of
Theorem
1.9
from
both
methods
which
are
essentially
global
in
nature
40
SHINICHI
MOCHIZUKI
[such
as
methods
involving
the
“global
address”
of
a
prime]
and
methods
which
are
essentially
local
in
nature
[such
as
methods
involving
p-adic
Hodge
theory
—
cf.
Remark
3.7.6,
(iii),
(v),
below].
This
point
of
view
concerning
the
“essential
independence
of
the
base
field”
is
developed
further
in
Remark
1.9.7
below.
Remark
1.9.6.
By
combining
the
theory
of
the
present
§1
with
the
theory
of
[Mzk21],
§1
[cf.,
e.g.,
[Mzk21],
Corollary
1.11
and
its
proof],
one
may
obtain
“functorial
group-theoretic
reconstruction
algorithms”,
in
a
number
of
cases,
for
finite
étale
coverings
of
configuration
spaces
associated
to
hyperbolic
curves.
We
leave
the
routine
details
to
the
interested
reader.
Remark
1.9.7.
One
way
to
think
of
the
construction
algorithm
in
Theorem
1.9
of
the
“NF-portion
of
the
function
field”
of
a
hyperbolic
orbicurve
of
strictly
Belyi
type
over
a
sub-p-adic
field
is
the
following:
The
algorithm
of
Theorem
1.9
may
be
thought
of
as
a
sort
of
complete
“combinatorialization”
—
independent
of
the
base
field!
—
of
the
[algebro-geometric
object
constituted
by
the]
orbicurve
under
considera-
tion.
This
sort
of
“combinatorialization”
may
be
thought
of
as
being
in
a
similar
vein
—
albeit
much
more
technically
complicated!
—
to
the
“combinatorialization”
of
a
category
of
finite
étale
coverings
of
a
connected
scheme
via
the
notion
of
an
abstract
Galois
category,
or
the
“combinatorialization”
of
certain
aspects
of
the
commutative
algebra
of
“normal
rings
with
toral
singularities”
via
the
abstract
monoids
that
appear
in
the
theory
of
log
regular
schemes
[cf.
also
the
Introduction
of
[Mzk16]
for
more
on
this
point
of
view].
Remark
1.9.8.
Typically
in
discussions
of
anabelian
geometry,
the
term
“group-
theoretic”
is
applied
to
a
property
or
construction
that
is
preserved
by
the
isomor-
phisms
[or
homomorphisms]
of
fundamental
groups
under
consideration
[cf.,
e.g.,
[Mzk5]].
By
contrast,
our
use
of
this
term
is
intended
in
a
stronger
sense.
That
is
to
say:
We
use
the
term
“group-theoretic
algorithm”
to
mean
that
the
algorithm
in
question
is
phrased
in
language
that
only
depends
on
the
topological
group
structure
of
the
fundamental
group
under
consideration.
[Thus,
the
more
“classical”
use
[e.g.,
in
[Mzk5]]
of
the
term
“group-theoretic”
cor-
responds,
in
our
discussion
of
“group-theoretic
algorithms”,
to
the
functoriality
—
e.g.,
with
respect
to
isomorphisms
of
some
type
—
of
the
algorithm.]
In
particu-
lar,
one
fundamental
difference
between
the
approach
usually
taken
to
anabelian
geometry
and
the
approach
taken
in
the
present
paper
is
the
following:
The
“classical”
approach
to
anabelian
geometry,
which
we
shall
refer
to
as
bi-anabelian,
centers
around
a
comparison
between
two
geomet-
ric
objects
[e.g.,
hyperbolic
orbicurves]
via
their
[arithmetic]
fundamental
TOPICS
IN
ABSOLUTE
ANABELIAN
GEOMETRY
III
41
groups.
By
contrast,
the
theory
of
the
present
paper,
which
we
shall
refer
to
as
mono-anabelian,
centers
around
the
task
of
establishing
“group-
theoretic
algorithms”
—
i.e.,
“group-theoretic
software”
—
that
require
as
input
data
only
the
[arithmetic]
fundamental
group
of
a
single
geometric
object.
Thus,
it
follows
formally
that
“mono-anabelian”
=⇒
“bi-anabelian”.
On
the
other
hand,
if
one
is
allowed
in
one’s
algorithms
to
introduce
some
fixed
reference
model
of
the
geometric
object
under
consideration,
then
the
task
of
establishing
an
“algorithm”
may,
in
effect,
be
reduced
to
“comparison
with
the
fixed
reference
model”,
i.e.,
reduced
to
some
sort
of
result
in
“bi-anabelian
geometry”.
That
is
to
say,
if
one
is
unable
to
settle
the
issue
of
ruling
out
the
use
of
such
models,
then
there
remains
the
possibility
that
?
“bi-anabelian”
=⇒
“mono-anabelian”.
We
shall
return
to
this
crucial
issue
in
§3
below
[cf.,
especially,
Remark
3.7.3].
Remark
1.9.9.
As
was
pointed
out
to
the
author
by
M.
Kim,
one
may
also
think
of
the
algorithms
of
a
result
such
as
Theorem
1.9
as
suggesting
an
approach
to
solving
the
problem
of
characterizing
“group-theoretically”
those
profinite
groups
Π
that
occur
[i.e.,
in
Theorem
1.9]
as
a
“Π
X
”.
That
is
to
say,
one
may
try
to
obtain
such
a
characterization
by
starting
with,
say,
an
arbitrary
slim
profinite
group
Π
and
then
proceeding
to
impose
“group-theoretic”
conditions
on
Π
corresponding
to
the
various
steps
of
the
algorithms
of
Theorems
1.9
—
i.e.,
conditions
whose
content
consists
of
minimal
assumptions
on
Π
that
are
necessary
in
order
to
execute
each
step
of
the
algorithm.
Corollary
1.10.
(Reconstruction
of
the
Function
Field
for
MLF’s)
Let
X
be
a
hyperbolic
orbicurve
over
an
MLF
k
[cf.
[Mzk20],
§0];
k
an
algebraic
closure
of
k;
k
NF
⊆
k
the
algebraic
closure
of
Q
in
k;
1
→
Δ
X
→
Π
X
→
G
k
→
1
def
def
—
where
Π
X
=
π
1
(X)
→
G
k
=
Gal(k/k)
denotes
the
natural
surjection
of
étale
fundamental
groups
[relative
to
some
choice
of
basepoints],
and
Δ
X
denotes
the
kernel
of
this
surjection
—
the
resulting
extension
of
profinite
groups.
Then:
(i)
There
exists
a
functorial
“group-theoretic”
algorithm
for
reconstruct-
∼
(G
k
))
→
Z
[cf.
(a)
below],
together
with
ing
the
natural
isomorphism
H
2
(G
k
,
μ
Z
∼
1
ab
[cf.
(b)
below]
from
the
(G
k
))
→
G
k
Z
the
natural
surjection
H
(G
k
,
μ
Z
profinite
group
G
k
,
as
follows:
(a)
Write:
def
ab
μ
Q/Z
(G
k
)
=
lim
−→
(H
)
tors
;
H
def
μ
(G
k
)
=
Hom(Q/Z,
μ
Q/Z
(G
k
))
Z
42
SHINICHI
MOCHIZUKI
—
where
H
ranges
over
the
open
subgroups
of
G
k
;
the
notation
“(−)
tors
”
denotes
the
torsion
subgroup
of
the
abelian
group
in
parentheses;
the
arrows
of
the
direct
limit
are
induced
by
the
Verlagerung,
or
transfer,
map
[cf.
the
discussion
preceding
[Mzk9],
Proposition
1.2.1;
the
proof
of
[Mzk9],
(G
k
)
is
Proposition
1.2.1].
[Thus,
the
underlying
module
of
μ
Q/Z
(G
k
),
μ
Z
unaffected
by
the
operation
of
passing
from
G
k
to
an
open
subgroup
of
G
k
.]
Then
one
constructs
the
natural
isomorphism
∼
H
2
(G
k
,
μ
(G
k
))
→
Z
Z
“group-theoretically”
from
G
k
via
the
algorithm
described
in
the
proof
of
[Mzk9],
Proposition
1.2.1,
(vii).
(b)
By
applying
the
isomorphism
of
(a)
[and
the
cup-product
in
group
coho-
mology],
one
constructs
the
surjection
∼
∼
unr
(G
k
))
→
G
ab
→
Z
H
1
(G
k
,
μ
k
G
Z
determined
by
the
Frobenius
element
in
the
maximal
unramified
quo-
tient
G
unr
of
G
k
via
the
“group-theoretic”
algorithm
described
in
the
proof
of
[Mzk9],
Proposition
1.2.1,
(ii),
(iv).
Here,
the
asserted
“functoriality”
is
with
respect
to
arbitrary
injective
open
homomorphisms
of
profinite
groups
[cf.
also
Remark
1.10.1,
(iii),
below].
(ii)
By
applying
the
functorial
“group-theoretic”
algorithm
of
[Mzk20],
Lemma
4.5,
(v),
to
construct
the
decomposition
groups
of
cusps
in
Π
X
,
one
ob-
(Π
X
)
as
in
Proposition
1.4,
(ii);
Theorem
1.9,
(b)
[cf.
also
tains
a
Π
X
-module
μ
Z
Remark
1.10.1,
(ii),
below].
Then
there
exists
a
functorial
“group-theoretic”
∼
algorithm
for
reconstructing
the
natural
isomorphism
μ
(G
k
)
→
μ
(Π
X
)
[cf.
Z
Z
(c)
below;
Remark
1.10.1
below]
and
the
image
of
a
certain
Kummer
map
[cf.
(d)
below]
from
the
profinite
group
Π
X
[cf.
Remark
1.9.2],
as
follows:
(c)
One
constructs
the
natural
isomorphism
[cf.,
e.g.,
[Mzk12],
Theorem
4.3]
∼
μ
(G
k
)
→
μ
(Π
X
)
Z
Z
—
thought
of
as
an
element
of
the
quotient
H
1
(G
k
,
μ
(Π
X
))
Hom(μ
(G
k
),
μ
(Π
X
))
Z
Z
Z
determined
by
the
surjection
of
(b)
—
as
the
unique
topological
genera-
tor
of
Hom(μ
(G
k
),
μ
(Π
X
))
that
is
contained
in
the
“positive
rational
Z
Z
structure”
[arising
from
various
J
ab
,
for
J
⊆
Δ
X
an
open
subgroup]
of
[Mzk9],
Lemma
2.5,
(i)
[cf.
also
[Mzk9],
Lemma
2.5,
(ii)].
(d)
One
constructs
the
image
of
the
Kummer
map
(Π
X
))
→
H
1
(Π
X
,
μ
(Π
X
))
k
×
→
H
1
(G
k
,
μ
Z
Z
TOPICS
IN
ABSOLUTE
ANABELIAN
GEOMETRY
III
43
as
the
inverse
image
of
the
subgroup
generated
by
the
Frobenius
element
∼
∼
(Π
X
))
→
H
1
(G
k
,
μ
(G
k
))
→
G
ab
via
the
surjection
H
1
(G
k
,
μ
k
Z
of
Z
Z
(b)
[cf.
also
the
isomorphism
of
(c)].
(d
)
Alternatively,
if
X
is
of
strictly
Belyi
type
[so
that
we
are
in
the
situ-
ation
of
Theorem
1.9],
then
one
may
construct
the
image
of
the
Kummer
map
of
(d)
—
without
applying
the
isomorphism
of
(c)
—
as
the
com-
×
(Π
))
k
pletion
of
H
1
(G
k
,
μ
X
NF
[cf.
Theorem
1.9,
(e)]
with
respect
to
Z
×
(Π
X
))
k
NF
)
{0}
[relative
to
the
the
valuation
on
the
field
(H
1
(G
k
,
μ
Z
additive
structure
of
Theorem
1.9,
(e)]
determined
by
the
subring
of
this
×
(Π
))
Z)
k
field
generated
by
the
intersection
Ker(H
1
(G
k
,
μ
X
NF
—
Z
×
,
where
“”
is
the
surjection
of
(b),
considered
up
to
multiplication
by
Z
an
object
which
is
independent
of
the
isomorphism
of
(c).
Here,
the
asserted
“functoriality”
is
with
respect
to
arbitrary
open
injective
homomorphisms
of
extensions
of
profinite
groups
—
cf.
Remark
1.10.1
below.
(iii)
Suppose
further
that
X
is
of
strictly
Belyi
type
[so
that
we
are
in
the
situation
of
Theorem
1.9].
Then
there
exists
a
functorial
“group-theoretic”
algorithm
for
reconstructing
the
function
field
K
X
of
X
from
the
profinite
group
Π
X
[cf.
Remark
1.9.2],
as
follows:
(e)
One
constructs
the
decomposition
groups
in
Π
X
of
arbitrary
closed
points
of
X
by
approximating
such
points
by
NF-points
of
X
[whose
decomposition
groups
have
already
been
constructed,
in
Theorem
1.9,
(a)],
via
the
equivalence
of
[Mzk12],
Lemma
3.1,
(i),
(iv).
(f)
For
S
a
finite
set
of
closed
points
of
X,
one
constructs
the
associated
“maximal
abelian
cuspidalization”
Π
c-ab
U
S
def
of
U
S
=
X\S
via
the
algorithm
of
[Mzk19],
Theorem
2.1,
(i)
[cf.
also
[Mzk19],
Theorem
1.1,
(iii),
as
well
as
Remark
1.10.4,
below].
Moreover,
by
applying
the
approximation
technique
of
(e)
to
the
Belyi
cuspidal-
izations
of
Theorem
1.9,
(a),
one
may
construct
the
Green’s
trivial-
izations
[cf.
[Mzk19],
Definition
2.1;
[Mzk19],
Remark
15]
for
arbitrary
pairs
of
closed
points
of
X
such
that
one
point
of
the
pair
is
an
NF-point;
in
particular,
one
may
construct
the
liftings
to
Π
c-ab
U
S
[from
Π
X
]
of
de-
composition
groups
of
NF-points.
(g)
By
applying
the
“maximal
abelian
cuspidalizations”
Π
c-ab
U
S
of
(f
),
together
with
the
characterization
of
principal
divisors
given
in
Proposition
1.6,
(ii)
[cf.
also
the
decomposition
groups
of
(e)],
one
constructs
the
subgroup
(Π
X
))
(
∼
(Π
X
)))
P
U
S
⊆
H
1
(Π
c-ab
=
H
1
(Π
U
S
,
μ
U
S
,
μ
Z
Z
[cf.
[Mzk19],
Proposition
2.1,
(i),
(ii)]
determined
by
the
cuspidal
prin-
cipal
divisors
via
the
isomorphisms
of
Theorem
1.9,
(b).
Then
the
44
SHINICHI
MOCHIZUKI
image
of
the
Kummer
map
in
P
U
S
may
be
constructed
as
the
collec-
(Π
X
))
—
where
tion
of
elements
of
P
U
S
whose
restriction
∈
H
1
(G
k
,
μ
Z
G
k
⊆
G
k
is
an
open
subgroup
corresponding
to
a
finite
extension
k
⊆
k
of
k
—
to
a
decomposition
group
of
some
NF-point
[cf.
(f
)]
is
contained
∼
in
(k
)
×
⊆
((k
)
×
)
∧
→
H
1
(G
k
,
μ
(Π
X
))
[cf.
(d)
or,
alternatively,(d’)].
Z
(h)
One
constructs
the
additive
structure
on
[the
image
—
cf.
(d)
—
of
]
k
×
{0}
as
the
unique
continuous
extension
of
the
additive
structure
×
on
(k
×
k
NF
)
{0}
constructed
in
Theorem
1.9,
(e).
One
constructs
the
image
of
the
Kummer
map
×
1
c-ab
K
X
→
lim
(Π
X
))
−→
H
(Π
U
S
,
μ
Z
S
by
letting
S
as
in
(g)
vary.
One
constructs
the
additive
structure
on
×
K
X
{0}
as
the
unique
additive
structure
compatible,
relative
to
the
oper-
ation
of
restriction
to
decomposition
groups
of
NF-points
[cf.
(f
)],
with
the
additive
structures
on
the
various
(k
)
×
{0},
for
k
⊆
k
a
finite
exten-
×
sion
of
k.
Also,
one
may
construct
the
restrictions
of
elements
of
K
X
to
decomposition
groups
not
only
of
NF-points,
but
also
of
arbitrary
closed
points
of
X,
by
approximating
as
in
(e);
this
allows
one
[by
letting
k
vary
among
finite
extensions
of
k
in
k]
to
give
an
alternative
construction
×
{0}
by
applying
Proposition
1.3
directly
of
the
additive
structure
on
K
X
[i.e.,
over
k,
as
opposed
to
k
NF
].
Here,
the
asserted
“functoriality”
is
with
respect
to
arbitrary
open
injective
homomorphisms
of
profinite
groups
[i.e.,
of
“Π
X
”]
—
cf.
Remark
1.10.1
below.
Proof.
The
validity
of
the
algorithms
asserted
in
Corollary
1.10
is
immediate
from
the
various
results
cited
in
the
statement
of
these
algorithms.
Remark
1.10.1.
(i)
In
general,
the
functoriality
of
Theorem
1.9,
Corollary
1.10,
when
applied
to
the
operation
of
passing
to
open
subgroups
of
Π
X
,
is
to
be
understood
in
the
sense
of
a
“compatibility”,
relative
to
dividing
the
usual
functorially
induced
morphism
(Π
X
)’s”
by
a
factor
given
by
the
index
of
the
subgroups
of
Δ
X
that
arise
on
“μ
Z
from
the
open
subgroups
of
Π
X
under
consideration
[cf.,
e.g.,
[Mzk19],
Remark
1].
(Π
U
)”
in
Theorem
1.9,
(b),
(ii)
In
fact,
strictly
speaking,
the
definition
of
“μ
Z
is
only
valid
if
U
is
a
hyperbolic
curve
of
genus
≥
2;
nevertheless,
one
may
extend
this
definition
to
the
case
where
U
is
an
arbitrary
hyperbolic
orbicurve
precisely
by
passing
to
coverings
and
applying
the
“functoriality/compatibility”
discussed
in
(i).
We
leave
the
routine
details
to
the
reader.
∼
(G
k
))
→
Z
of
(iii)
In
a
similar
vein,
note
that
the
isomorphism
H
2
(G
k
,
μ
Z
Corollary
1.10,
(a),
is
functorial
in
the
sense
that
it
is
compatible
with
the
result
of
dividing
the
usual
functorially
induced
morphism
by
a
factor
given
by
the
index
of
the
open
subgroups
of
G
k
under
consideration.
TOPICS
IN
ABSOLUTE
ANABELIAN
GEOMETRY
III
45
Remark
1.10.2.
Just
as
was
the
case
with
Theorem
1.9,
one
may
give
a
tempered
version
of
Corollary
1.10
—
cf.
Remark
1.9.1.
Remark
1.10.3.
(i)
The
isomorphism
of
Corollary
1.10,
(c),
may
be
thought
of
as
a
sort
of
“synchronization
of
[arithmetic
and
geometric]
cyclotomes”,
in
the
style
of
the
“synchronization
of
cyclotomes”
given
in
the
final
display
of
Proposition
1.4,
(ii).
(ii)
One
may
construct
the
natural
isomorphism
∼
1
(Π
X
))
G
ab
k
→
H
(G
k
,
μ
Z
by
applying
the
displayed
isomorphism
of
Corollary
1.10,
(c),
to
the
inverse
of
the
first
displayed
isomorphism
of
Corollary
1.10,
(b).
By
applying
this
natural
isomor-
phism
to
various
open
subgroups
of
G
k
,
we
thus
obtain
yet
another
isomorphism
of
cyclotomes
×
∼
def
κ
μ
(G
)
→
μ
(Π
)
=
Hom(Q/Z,
κ(k
k
X
NF
))
Z
Z
×
×
—
where
we
write
κ(k
NF
)
for
the
image
of
k
NF
in
1
lim
(Π
U
))
−→
H
(Π
V
,
μ
Z
V
via
the
inclusion
induced
by
the
Kummer
map
in
the
display
of
Theorem
1.9,
(d).
Remark
1.10.4.
Here,
we
take
the
opportunity
to
correct
an
unfortunate
misprint
→
X,
Z
Y
→
Y
in
the
proof
of
[Mzk19],
Theorem
1.1,
(iii).
The
phrase
“Z
X
are
diagonal
coverings”
that
appears
at
the
beginning
of
this
proof
should
read
→
X
×
X,
Z
Y
→
Y
×
Y
are
diagonal
coverings”.
“Z
X
Finally,
we
conclude
the
present
§1
by
observing
that
the
techniques
developed
in
the
present
§1
may
be
intepreted
as
implying
a
very
elementary
semi-absolute
birational
analogue
of
Theorem
1.9.
Theorem
1.11.
(Semi-absolute
Reconstruction
of
Function
Fields
of
Curves
over
Kummer-faithful
Fields)
Let
X
be
a
smooth,
proper,
geometrically
connected
curve
of
genus
g
X
over
a
Kummer-faithful
field
k;
K
X
the
function
def
field
of
X;
η
X
=
Spec(K
X
);
k
an
algebraic
closure
of
k;
1
→
Δ
η
X
→
Π
η
X
→
G
k
→
1
def
def
—
where
Π
η
X
=
π
1
(η
X
)
→
G
k
=
Gal(k/k)
denotes
the
natural
surjection
of
étale
fundamental
groups
[relative
to
some
choice
of
basepoints],
and
Δ
η
X
denotes
the
kernel
of
this
surjection
—
the
resulting
extension
of
profinite
groups.
Then
Δ
η
X
,
Π
η
X
,
and
G
k
are
slim.
For
simplicity,
let
us
suppose
further
[for
instance,
by
replacing
X
by
a
finite
étale
covering
of
X]
that
g
X
≥
2.
Then
there
exists
a
functorial
“group-theoretic”
algorithm
for
reconstructing
the
function
field
46
SHINICHI
MOCHIZUKI
K
X
from
the
extension
of
profinite
groups
1
→
Δ
η
X
→
Π
η
X
→
G
k
→
1;
this
algorithm
consists
of
the
following
steps:
(a)
Let
l
be
a
prime
number.
If
ρ
:
G
k
→
Z
×
l
is
a
character,
and
M
is
an
abelian
pro-l
group
equipped
with
a
continuous
action
by
G
k
,
then
let
us
write
F
ρ
(M
)
⊆
M
for
the
closed
subgroup
topologically
generated
by
the
closed
subgroups
of
M
that
are
isomorphic
to
Z
l
(ρ)
[i.e.,
the
G
k
-
module
obtained
by
letting
G
k
act
on
Z
l
via
ρ]
as
H-modules,
for
some
open
subgroup
H
⊆
G
k
.
[Thus,
F
ρ
(M
)
⊆
M
depends
only
on
the
“power-
equivalence
class”
of
ρ
—
cf.
Remark
1.5.1.]
Then
the
power-equivalence
class
of
the
cyclotomic
character
χ
l
:
G
k
→
Z
×
l
may
be
characterized
⊗
Z
)
is
not
topologically
finitely
generated.
by
the
condition
that
F
χ
l
(Δ
ab
l
η
X
(b)
Let
l
be
a
prime
number.
If
M
is
an
abelian
pro-l
group
equipped
with
a
continuous
action
by
G
k
such
that
M/F
χ
l
(M
)
is
topologically
finitely
generated,
then
let
us
write
M
T
(M
)
for
the
maximal
torsion-
free
quasi-trivial
quotient
[i.e.,
maximal
torsion-free
quotient
on
which
G
k
acts
through
a
finite
quotient].
[Thus,
one
verifies
immediately
that
“T
(M
)”
is
well-defined.]
Then
one
may
compute
the
genus
of
X
via
the
formula
[cf.
the
proof
of
[Mzk21],
Corollary
2.10]
ab
2g
X
=
dim
Q
l
(Q(Δ
ab
η
X
⊗
Z
l
)
⊗
Q
l
)
+
dim
Q
l
(T
(Δ
η
X
⊗
Z
l
)
⊗
Q
l
)
def
—
where
we
write
Q(−)
=
(−)/F
χ
l
(−).
In
particular,
this
allows
one
to
characterize,
via
the
Hurwitz
formula,
those
pairs
of
open
subgroups
J
i
⊆
H
i
⊆
Δ
η
X
such
that
“the
covering
between
J
i
and
H
i
is
cyclic
of
or-
der
a
power
of
l
and
totally
ramified
at
precisely
one
closed
point
but
un-
ramified
elsewhere”
[cf.
the
proof
of
[Mzk21],
Corollary
2.10].
Moreover,
this
last
characterization
implies
a
“group-theoretic”
characterization
of
the
inertia
subgroups
I
x
⊆
Δ
η
X
of
points
x
∈
X(k)
[cf.
the
proof
of
[Mzk21],
Corollary
2.10;
the
latter
portion
of
the
proof
of
[Mzk9],
Lemma
1.3.9],
hence
of
the
quotient
Δ
η
X
Δ
X
[whose
kernel
is
topologically
normally
generated
by
the
I
x
,
for
x
∈
X(k)].
Finally,
the
decomposition
group
D
x
⊆
Π
η
X
of
x
∈
X(k)
may
then
be
constructed
as
the
normal-
izer
[or,
equivalently,
commensurator]
of
I
x
in
Π
η
X
[cf.,
e.g.,
[Mzk12],
Theorem
1.3,
(ii)].
∼
(c)
One
may
construct
the
natural
isomorphisms
I
x
→
M
X
[where
x
∈
X(k);
M
X
is
as
in
Proposition
1.4,
(ii)]
via
the
technique
of
Proposition
1.4,
(ii).
These
isomorphisms
determine
[by
restriction
to
the
I
x
]
a
natural
map
Z
H
1
(Π
η
X
,
M
X
)
→
x∈X(k)
[cf.
Proposition
1.6,
(iii)].
Denote
by
P
η
X
⊆
H
1
(Π
η
X
,
M
X
)
[cf.
Proposi-
tion
1.8]
the
inverse
image
in
H
1
(Π
η
X
,
M
X
)
of
the
subgroup
of
x∈X(k)
Z
TOPICS
IN
ABSOLUTE
ANABELIAN
GEOMETRY
III
47
consisting
of
the
principal
divisors
—
i.e.,
divisors
D
of
degree
zero
supported
on
a
collection
of
points
∈
X(k
H
),
where
k
H
⊆
k
is
the
subfield
corresponding
to
an
open
subgroup
H
⊆
G
k
,
whose
associated
class
∈
H
1
(H,
Δ
ab
X
)
[i.e.,
the
class
obtained
as
the
difference
between
the
section
“t
D
”
of
Proposition
1.6,
(ii),
and
the
identity
section]
is
trivial.
(d)
The
image
of
the
Kummer
map
×
K
X
→
H
1
(Π
η
X
,
M
X
)
may
be
constructed
as
the
subgroup
generated
by
elements
θ
∈
P
η
X
for
which
there
exists
an
x
∈
X(k)
such
that
θ|
x
∈
H
1
(D
x
,
M
X
)
vanishes
[i.e.,
=
1,
if
one
works
multiplicatively]
—
cf.
the
technique
of
Proposition
×
{0}
may
be
recovered
1.8,
(i).
Moreover,
the
additive
structure
on
K
X
via
the
algorithm
of
Proposition
1.3.
Finally,
the
asserted
“functoriality”
is
with
respect
to
arbitrary
open
injective
homomorphisms
of
extensions
of
profinite
groups
[cf.
Remark
1.10.1,
(i)].
Proof.
The
slimness
of
Δ
η
X
follows
immediately
from
the
argument
applied
to
verify
the
slimness
portion
of
[Mzk21],
Corollary
2.10.
The
validity
of
the
recon-
struction
algorithm
asserted
in
Theorem
1.11
is
immediate
from
the
various
results
cited
in
the
statement
of
this
algorithm.
Now,
by
applying
the
functoriality
of
this
algorithm,
the
slimness
of
G
k
follows
immediately
from
the
argument
applied
in
[Mzk5],
Lemma
15.8,
to
verify
the
slimness
of
G
k
when
k
is
sub-p-adic.
Finally,
the
slimness
of
Π
η
X
follows
from
the
slimness
of
Δ
η
X
,
G
k
.
Remark
1.11.1.
(i)
One
verifies
immediately
that
when
k
is
an
MLF,
the
semi-absolute
algo-
rithms
of
Theorem
1.11
may
be
rendered
absolute
[i.e.,
one
may
construct
the
kernel
of
the
quotient
“Π
η
X
G
k
”]
by
applying
the
algorithm
that
is
implicit
in
the
proof
of
the
corresponding
portion
of
[Mzk21],
Corollary
2.10.
(ii)
Suppose,
in
the
notation
of
Theorem
1.11
that
k
is
an
NF.
Then
an
absolute
version
of
the
functoriality
portion
[i.e.,
the
“Grothendieck
Conjecture”
portion]
of
Theorem
1.11
is
proven
in
[Pop]
[cf.
[Pop],
Theorem
2].
Moreover,
in
[Pop],
Observation
[and
the
following
discussion],
an
algorithm
is
given
for
passing
from
the
absolute
data
“Π
η
X
”
to
the
semi-absolute
data
“(Π
η
X
,
Δ
η
X
⊆
Π
η
X
)”.
Thus,
by
combining
this
algorithm
of
[Pop]
with
Theorem
1.11,
one
obtains
an
absolute
version
of
Theorem
1.11.
Remark
1.11.2.
One
may
think
of
the
argument
used
to
prove
the
slimness
of
G
k
in
the
proof
of
Theorem
1.11
[i.e.,
the
argument
of
the
proof
of
[Mzk5],
Lemma
15.8]
as
being
similar
in
spirit
to
the
proof
[cf.,
e.g.,
[Mzk9],
Theorem
1.1.1,
(ii)]
of
the
slimness
of
G
k
via
local
class
field
theory
in
the
case
where
k
is
an
MLF,
as
well
as
to
the
proof
of
the
slimness
of
the
geometric
fundamental
group
of
a
hyperbolic
curve
given,
for
instance,
in
[MT],
Proposition
1.4,
via
the
induced
action
on
the
48
SHINICHI
MOCHIZUKI
torsion
points
of
the
Jacobian
of
the
curve,
in
which
the
curve
may
be
embedded.
That
is
to
say,
in
the
case
where
k
is
an
arbitrary
Kummer-faithful
field,
since
one
does
not
have
an
analogue
of
local
class
field
theory
(respectively,
of
the
embedding
of
a
curve
in
its
Jacobian),
the
moduli
of
hyperbolic
curves
over
k,
in
the
context
of
a
relative
anabelian
result
for
the
arithmetic
fundamental
groups
of
such
curves,
plays
the
role
of
the
abelianization
of
G
k
(respectively,
of
the
torsion
points
of
the
Jacobian)
in
the
case
where
k
is
an
MLF
(respectively,
in
the
case
of
the
geometric
fundamental
group
of
a
hyperbolic
curve)
—
i.e.,
the
role
of
a
“functorial,
group-
theoretically
reconstructible
embedding”
of
k
(respectively,
the
curve).
Remark
1.11.3.
It
is
interesting
to
note
that
the
techniques
that
appear
in
the
algorithms
of
Theorem
1.11
are
extremely
elementary.
For
instance,
unlike
the
case
with
Theorem
1.9,
Corollary
1.10,
the
algorithms
of
Theorem
1.11
do
not
depend
on
the
somewhat
difficult
[e.g.,
in
their
use
of
p-adic
Hodge
theory]
results
of
[Mzk5].
Put
another
way,
this
elementary
nature
of
Theorem
1.11
serves
to
highlight
the
fact
that
the
only
non-elementary
portion
[in
the
sense
of
its
dependence
of
the
results
of
[Mzk5]]
of
the
algorithms
of
Theorem
1.9
is
the
use
of
the
technique
of
Belyi
cuspidalizations.
It
is
precisely
this
“non-elementary
portion”
of
Theorem
1.9
that
requires
us,
in
Theorem
1.9,
to
assume
that
the
base
field
is
sub-p-adic
[i.e.,
as
opposed
to
merely
Kummer-faithful,
as
in
Theorem
1.11].
Remark
1.11.4.
(i)
The
observation
of
Remark
1.11.3
prompts
the
following
question:
If
the
birational
version
[i.e.,
Theorem
1.11]
of
Theorem
1.9
is
so
much
more
elementary
than
Theorem
1.9,
then
what
is
the
advantage
[i.e.,
relative
to
the
anabelian
geometry
of
function
fields]
of
considering
the
anabelian
geometry
of
hyperbolic
curves?
One
key
advantage
of
working
with
hyperbolic
curves,
in
the
context
of
the
theory
of
the
present
paper,
lies
in
the
fact
that
“most”
hyperbolic
curves
admit
a
core
[cf.
[Mzk3],
§3;
[Mzk10],
§2].
Moreover,
the
existence
of
“cores”
at
the
level
of
schemes
has
a
tendency
to
imply
to
existence
of
“cores”
at
the
level
of
“étale
fundamental
groups
considered
geometrically”,
i.e.,
at
the
level
of
anabelioids
[cf.
[Mzk11],
§3.1].
The
existence
of
a
core
is
crucial
to,
for
instance,
the
theory
of
the
étale
theta
function
given
in
[Mzk18],
§1,
§2,
and,
moreover,
in
the
present
three-part
series,
plays
an
important
role
in
the
theory
of
elliptically
admissible
[cf.
[Mzk21],
Definition
3.1]
hyperbolic
orbicurves.
On
the
other
hand,
it
is
easy
to
see
that
“function
fields
do
not
admit
cores”:
i.e.,
if,
in
the
notation
of
Theorem
1.11,
we
write
Loc(η
X
)
for
the
category
whose
objects
are
connected
schemes
that
admit
a
connected
finite
étale
covering
which
is
also
a
connected
finite
étale
covering
of
η
X
,
and
whose
morphisms
are
the
finite
étale
morphisms,
then
Loc(η
X
)
fails
to
admit
a
terminal
object.
(ii)
The
observation
of
(i)
is
interesting
in
the
context
of
the
theory
of
§5
below,
in
which
we
apply
various
[mono-]anabelian
results
to
construct
“canonical
rigid
integral
structures”
called
“log-shells”.
Indeed,
in
the
Introduction
to
TOPICS
IN
ABSOLUTE
ANABELIAN
GEOMETRY
III
49
[Mzk11],
it
is
explained,
via
analogy
to
the
complex
analytic
theory
of
the
upper
half-plane,
how
the
notion
of
a
core
may
be
thought
of
as
a
sort
of
“canonical
integral
structure”
—
i.e.,
relative
to
the
“modifications
of
integral
structure”
constituted
by
“going
up
and
down
via
various
finite
étale
coverings”.
Here,
it
is
interesting
to
note
that
this
idea
of
a
“canonical
integral
structure
relative
to
going
up
and
down
via
finite
étale
coverings”
may
also
be
seen
in
the
theory
surrounding
the
property
of
cyclotomic
rigidity
in
the
context
of
the
étale
theta
function
[cf.,
e.g.,
[Mzk18],
Remark
2.19.3].
Moreover,
let
us
observe
that
these
“integral
structures
with
respect
to
finite
étale
coverings”
may
be
thought
of
as
“exponentiated
integral
structures”
—
i.e.,
in
the
sense
that,
for
instance,
in
the
case
of
G
m
over
Q,
these
integral
structures
are
not
integral
structures
relative
to
the
scheme-theoretic
base
ring
Z
⊆
Q,
but
rather
with
respect
to
the
exponent
of
the
standard
coordinate
U
,
which,
via
multiplication
by
various
nonnegative
integers
N
,
gives
rise,
in
the
form
of
mappings
U
n
→
U
N
·n
,
to
various
finite
étale
coverings
of
G
m
.
Such
“non-scheme-
theoretic
exponentiated
copies
of
Z”
play
an
important
role
in
the
theory
of
the
étale
theta
function
as
the
Galois
group
of
a
certain
natural
infinite
étale
covering
of
the
Tate
curve
—
cf.
the
discussion
of
[Mzk18],
Remark
2.16.2.
Moreover,
the
idea
of
constructing
“canonical
integral
structures”
by
“de-exponentiating
certain
exponentiated
integral
structures”
may
be
rephrased
as
the
idea
of
“constructing
canonical
integral
structures
by
applying
some
sort
of
logarithm
operation”.
From
this
point
of
view,
such
“canonical
integral
structures
with
respect
to
finite
étale
coverings”
are
quite
reminiscent
of
the
canonical
integral
structures
arising
from
log-shells
to
be
constructed
in
§5
below.
Remark
1.11.5.
In
the
context
of
the
discussion
of
Remark
1.11.4,
if
the
hyper-
bolic
curve
in
question
is
affine,
then,
relative
to
the
function
field
of
the
curve,
the
additional
data
necessary
to
determine
the
given
affine
hyperbolic
curve
consists
precisely
of
some
[nonempty]
finite
collection
of
conjugacy
classes
of
inertia
groups
[i.e.,
“I
x
”
as
in
Theorem
11.1,
(b)].
Thus,
from
the
point
of
view
of
the
discussion
of
Remark
1.11.3,
the
technique
of
Belyi
cuspidalizations
is
applied
precisely
so
as
to
enable
one
to
work
with
this
additional
data
[cf.
also
the
discussion
of
Remark
3.7.7
below].
50
SHINICHI
MOCHIZUKI
Section
2:
Archimedean
Reconstruction
Algorithms
In
the
present
§2,
we
re-examine
various
aspects
of
the
complex
analytic
the-
ory
of
[Mzk14]
from
an
algorithm-based,
“model-implicit”
[cf.
Remark
2.7.4
below]
point
of
view
motivated
by
the
Galois-theoretic
theory
of
§1.
More
precisely,
the
“SL
2
(R)-based
approach”
of
[Mzk14],
§1,
may
be
seen
in
the
general
theory
of
Aut-holomorphic
spaces
given
in
Proposition
2.2,
Corollary
2.3,
while
the
“paral-
lelograms,
rectangles,
squares
approach”
of
[Mzk14],
§2,
is
developed
further
in
the
reconstruction
algorithms
of
Propositions
2.5,
2.6.
These
two
approaches
are
com-
bined
to
obtain
the
main
result
of
the
present
§2
[cf.
Corollary
2.7],
which
consists
of
a
certain
reconstruction
algorithm
for
the
“local
linear
holomorphic
structure”
of
an
Aut-holomorphic
orbispace
arising
from
an
elliptically
admissible
hyperbolic
orbicurve.
Finally,
in
Corollaries
2.8,
2.9,
we
consider
the
relationship
between
Corollary
2.7
and
the
global
portion
of
the
Galois-theoretic
theory
of
§1.
The
following
definition
will
play
an
important
role
in
the
theory
of
the
present
§2.
Definition
2.1.
(i)
Let
X
be
a
Riemann
surface
[i.e.,
a
complex
manifold
of
dimension
one].
Write
A
X
for
the
assignment
that
assigns
to
each
connected
open
subset
U
⊆
X
the
group
def
A
X
(U
)
=
Aut
hol
(U
)
of
holomorphic
automorphisms
of
U
—
which
we
think
of
as
being
“some
distin-
guished
subgroup”
of
the
group
of
self-homeomorphisms
Aut(U
top
)
of
the
under-
lying
topological
space
U
top
of
U
.
We
shall
refer
to
as
the
Aut-holomorphic
space
associated
to
X
the
pair
def
X
=
(X
top
,
A
X
)
def
consisting
of
the
underlying
topological
space
X
top
=
X
top
of
X,
together
with
the
def
assignment
A
X
=
A
X
;
also,
we
shall
refer
to
the
assignment
A
X
=
A
X
as
the
Aut-
holomorphic
structure
on
X
top
=
X
top
[determined
by
X].
If
X
is
biholomorphic
to
the
open
unit
disc,
then
we
shall
refer
to
X
as
an
Aut-holomorphic
disc.
If
X
is
a
hyperbolic
Riemann
surface
of
finite
type
(respectively,
a
hyperbolic
Riemann
surface
of
finite
type
associated
to
an
elliptically
admissible
[cf.
[Mzk21],
Definition
3.1]
hyperbolic
curve
over
C),
then
we
shall
refer
to
the
Aut-holomorphic
space
X
as
hyperbolic
of
finite
type
(respectively,
elliptically
admissible).
If
U
is
a
collection
of
connected
open
subsets
of
X
that
forms
a
basis
for
the
topology
of
X
and,
moreover,
satisfies
the
condition
that
any
connected
open
subset
of
X
that
is
contained
in
an
element
of
U
is
itself
an
element
of
U
,
then
we
shall
refer
to
U
as
a
local
structure
on
X
top
and
to
the
restriction
A
X
|
U
of
A
X
to
U
as
a
[U-local]
pre-Aut-holomorphic
structure
on
X
top
.
(ii)
Let
X
(respectively,
Y
)
be
a
Riemann
surface;
X
(respectively,
Y)
the
Aut-holomorphic
space
associated
to
X
(respectively,
Y
);
U
(respectively,
V)
a
local
structure
on
X
top
(respectively,
Y
top
).
Then
we
shall
refer
to
as
a
(U,
V)-local
morphism
of
Aut-holomorphic
spaces
φ
:
X
→
Y
TOPICS
IN
ABSOLUTE
ANABELIAN
GEOMETRY
III
51
any
local
isomorphism
of
topological
spaces
φ
top
:
X
top
→
Y
top
with
the
property
that
for
any
open
subset
U
X
∈
U
that
maps
homeomorphically
via
φ
top
onto
some
∼
open
subset
U
Y
∈
V,
φ
top
induces
a
bijection
A
X
(U
X
)
→
A
Y
(U
Y
);
when
U,
V
are,
respectively,
the
sets
of
all
connected
open
subsets
of
X,
Y
,
then
we
shall
omit
the
word
“(U
,
V)-local”
from
this
terminology;
when
φ
top
is
a
finite
covering
space
map,
we
shall
say
that
φ
is
finite
étale.
We
shall
refer
to
a
map
X
→
Y
which
is
either
holomorphic
or
anti-holomorphic
at
each
point
of
X
as
an
RC-holomorphic
morphism
[cf.
[Mzk14],
Definition
1.1,
(vi)].
(iii)
Let
Z,
Z
be
orientable
topological
surfaces
[i.e.,
two-manifolds].
If
p
∈
Z,
then
let
us
write
def
ab
Orn(Z,
p)
=
lim
−→
π
1
(W
\{p})
W
—
where
W
ranges
over
the
connected
open
neighborhoods
of
p
in
Z;
“π
1
(−)”
denotes
the
usual
topological
fundamental
group,
relative
to
some
basepoint
[so
“π
1
(−)”
is
only
defined
up
to
inner
automorphisms,
an
indeterminacy
which
may
be
eliminated
by
passing
to
the
abelianization
“ab”];
thus,
Orn(Z,
p)
is
[noncanon-
ically!]
isomorphic
to
Z.
Note
that
since
Z
is
orientable,
it
follows
that
the
assign-
ment
p
→
Orn(Z,
p)
determines
a
trivial
local
system
on
Z,
whose
module
of
global
sections
we
shall
denote
by
Orn(Z)
[so
Orn(Z)
is
a
direct
product
of
copies
of
Z,
in-
dexed
by
the
connected
components
of
Z].
One
verifies
immediately
that
any
local
isomorphism
Z
→
Z
induces
a
well-defined
homomorphism
Orn(Z)
→
Orn(Z
).
We
shall
say
that
any
two
local
isomorphisms
α,
β
:
Z
→
Z
are
co-oriented
if
they
induce
the
same
homomorphism
Orn(Z)
→
Orn(Z
).
We
shall
refer
to
as
a
pre-co-orientation
ζ
:
Z
→
Z
any
equivalence
class
of
local
isomorphisms
Z
→
Z
relative
to
the
equivalence
relation
determined
by
the
property
of
being
co-oriented
[so
a
pre-co-orientation
may
be
thought
of
as
a
collection
of
maps
Z
→
Z
,
or,
al-
ternatively,
as
a
homomorphism
Orn(Z)
→
Orn(Z
)].
Thus,
the
pre-co-orientations
from
the
open
subsets
of
Z
to
Z
form
a
pre-sheaf
on
Z;
we
shall
refer
to
as
a
co-orientation
ζ
:
Z
→
Z
any
section
of
the
sheafification
of
this
pre-sheaf
[so
a
co-orientation
may
be
thought
of
as
a
collection
of
maps
from
open
subsets
of
Z
to
Z
,
or,
alternatively,
as
a
homomorphism
Orn(Z)
→
Orn(Z
)].
(iv)
Let
X,
Y
,
X,
Y,
U,
V
be
as
in
(ii).
Then
we
shall
say
that
two
(U,
V)-local
morphisms
of
Aut-holomorphic
spaces
φ
1
,
φ
2
:
X
→
Y
are
co-holomorphic
if
φ
top
1
and
φ
top
are
co-oriented
[cf.
(iii)].
We
shall
refer
to
as
a
pre-co-holomorphicization
2
ζ
:
X
→
Y
any
equivalence
class
of
(U,
V)-local
morphisms
of
Aut-holomorphic
spaces
X
→
Y
relative
to
the
equivalence
relation
determined
by
the
property
of
being
co-holomorphic
[so
a
pre-co-holomorphicization
may
be
thought
of
as
a
collection
of
maps
from
X
top
to
Y
top
].
Thus,
the
pre-co-holomorphicizations
from
the
Aut-holomorphic
spaces
determined
by
open
subsets
of
X
top
to
Y
form
a
pre-
sheaf
on
X
top
;
we
shall
refer
to
as
a
co-holomorphicization
[cf.
also
Remark
2.3.2
below]
ζ
:
X
→
Y
any
section
of
the
sheafification
of
this
pre-sheaf
[so
a
co-holomorphicization
may
be
thought
of
as
a
collection
of
maps
from
open
subsets
of
X
top
to
Y
top
].
Finally,
we
52
SHINICHI
MOCHIZUKI
observe
that
every
co-holomorphicization
(respectively,
pre-co-holomorphicization)
determines
a
co-orientation
(pre-co-orientation)
between
the
underlying
topological
spaces.
Remark
2.1.1.
One
verifies
immediately
that
there
is
a
natural
extension
of
the
notions
of
Definition
2.1
to
the
case
of
Riemann
orbisurfaces,
which
give
rise
to
“Aut-holomorphic
orbispaces”
[not
to
be
confused
with
the
“orbi-objects”
of
§0,
which
will
always
be
identifiable
in
the
present
paper
by
means
of
the
hyphen
“-”
following
the
prefix
“orbi”].
Here,
we
understand
the
term
“Riemann
orbisurface”
to
refer
to
a
one-dimensional
complex
analytic
stack
which
is
locally
isomorphic
to
the
complex
analytic
stack
obtained
by
forming
the
stack-theoretic
quotient
of
a
Riemann
surface
[i.e.,
a
one-dimensional
complex
manifold]
by
a
finite
group
of
[holomorphic]
automorphisms
[of
the
Riemann
surface].
In
particular,
a
“Riemann
orbiface”
is
necessarily
a
Riemann
surface
over
the
complement,
in
the
“coarse
space”
associated
to
the
orbisurface,
of
some
discrete
closed
subset.
Remark
2.1.2.
One
important
aspect
of
the
“Aut-holomorphic”
approach
to
the
notion
of
a
“holomorphic
structure”
is
that
this
approach
has
the
virtue
of
being
free
of
any
mention
of
some
“fixed
reference
model”
copy
of
the
field
of
complex
numbers
C
—
cf.
Remark
2.7.4
below.
Proposition
2.2.
(Commensurable
Terminality
of
RC-Holomorphic
Automorphisms
of
the
Disc)
Let
X,
Y
be
Aut-holomorphic
discs,
arising,
respectively,
from
Riemann
surfaces
X,
Y
.
Then:
∼
(i)
Every
isomorphism
of
Aut-holomorphic
spaces
X
→
Y
arises
from
a
∼
unique
RC-holomorphic
isomorphism
X
→
Y
.
(ii)
Let
us
regard
the
group
Aut(X
top
)
as
equipped
with
the
compact-open
topology.
Then
the
subgroup
Aut
RC-hol
(X)
⊆
Aut(X
top
)
of
RC-holomorphic
automorphisms
of
X,
which
[as
is
well-known]
contains
Aut
hol
(X)
as
a
subgroup
of
index
two,
is
closed
and
commensurably
terminal
[cf.
[Mzk20],
§0].
Moreover,
we
have
isomorphisms
of
topological
groups
Aut
hol
(X)
∼
=
SL
2
(R)/{±1};
Aut
RC-hol
(X)
∼
=
GL
2
(R)/R
×
[where
we
regard
Aut
hol
(X),
Aut
RC-hol
(X),
as
equipped
with
the
topology
induced
by
the
topology
of
Aut(X
top
),
i.e.,
the
compact-open
topology].
Proof.
It
is
immediate
from
the
definitions
that
assertion
(i)
follows
formally
from
the
commensurable
terminality
[in
fact,
in
this
situation,
normal
terminality
suffices]
of
assertion
(ii).
Thus,
it
suffices
to
verify
assertion
(ii).
First,
we
recall
that
we
have
a
natural
isomorphism
of
connected
topological
groups
Aut
hol
(X)
∼
=
SL
2
(R)/{±1}
TOPICS
IN
ABSOLUTE
ANABELIAN
GEOMETRY
III
53
[where
we
regard
Aut
hol
(X)
as
equipped
with
the
compact-open
topology].
Next,
let
us
recall
the
well-known
fact
in
elementary
complex
analysis
that
“a
sequence
of
holomorphic
functions
on
X
top
that
converges
uniformly
on
compact
subsets
of
X
top
converges
to
a
holomorphic
function
on
X
top
”.
[This
fact
is
often
ap-
plied
in
proofs
of
the
Riemann
mapping
theorem.]
This
fact
implies
immedi-
ately
that
Aut
hol
(X),
Aut
RC-hol
(X)
are
closed
in
Aut(X
top
).
Now
suppose
that
α
∈
Aut(X
top
)
lies
in
the
commensurator
of
Aut
hol
(X);
thus,
the
intersection
(α
·
Aut
hol
(X)·α
−1
)
Aut
hol
(X)
is
a
closed
subgroup
of
finite
index
of
Aut
hol
(X).
But
this
implies
that
(α·Aut
hol
(X)·α
−1
)
Aut
hol
(X)
is
an
open
subgroup
of
Aut
hol
(X),
hence
[since
Aut
hol
(X)
is
connected]
that
(α
·
Aut
hol
(X)
·
α
−1
)
Aut
hol
(X)
=
Aut
hol
(X),
i.e.,
that
α
·
Aut
hol
(X)
·
α
−1
⊇
Aut
hol
(X).
Thus,
by
replacing
α
by
α
−1
,
we
conclude
that
α
normalizes
Aut
hol
(X),
i.e.,
that
α
induces
an
automor-
phism
of
the
topological
group
Aut
hol
(X)
∼
=
SL
2
(R)/{±1},
hence
also
[by
Cartan’s
theorem
—
cf.,
e.g.,
[Serre],
Chapter
V,
§9,
Theorem
2;
the
proof
of
[Mzk14],
Lemma
1.10]
of
the
real
analytic
Lie
group
SL
2
(R)/{±1}.
Thus,
as
is
well-known,
it
fol-
lows
[for
instance,
by
considering
the
action
of
α
on
the
Borel
subalgebras
of
the
complexification
of
the
Lie
algebra
of
SL
2
(R)/{±1}]
that
α
arises
from
an
element
of
GL
2
(C)/C
×
that
fixes
[relative
to
the
action
by
conjugation]
the
Lie
subalgebra
sl
2
(R)
of
sl
2
(C).
But
such
an
element
of
GL
2
(C)/C
×
is
easily
verified
to
be
an
element
of
GL
2
(R)/R
×
.
In
particular,
by
considering
the
action
of
α
on
maximal
compact
subgroups
of
Aut
hol
(X)
[cf.
the
proof
of
[Mzk14],
Lemma
1.10],
it
follows
that
α
arises
from
an
RC-holomorphic
automorphism
of
X,
as
desired.
In
fact,
as
the
following
result
shows,
the
notions
of
an
Aut-holomorphic
struc-
ture
and
a
pre-Aut-holomorphic
structure
are
equivalent
to
one
another,
as
well
as
to
the
usual
notion
of
a
“holomorphic
structure”.
Corollary
2.3.
(Morphisms
of
Aut-Holomorphic
Spaces)
Let
X
(respec-
tively,
Y
)
be
a
Riemann
surface;
X
(respectively,
Y)
the
Aut-holomorphic
space
associated
to
X
(respectively,
Y
);
U
(respectively,
V)
a
local
structure
on
X
top
(re-
spectively,
Y
top
).
Then:
(i)
Every
(U
,
V)-local
morphism
of
Aut-holomorphic
spaces
φ
:
X
→
Y
arises
from
a
unique
étale
RC-holomorphic
morphism
ψ
:
X
→
Y
.
Moreover,
if,
in
this
situation,
X,
Y
[i.e.,
X
top
,
Y
top
]
are
connected,
then
there
exist
pre-
cisely
two
co-holomorphicizations
X
→
Y,
corresponding
to
the
holomorphic
and
anti-holomorphic
local
isomorphisms
from
open
subsets
of
X
to
Y
.
(ii)
Every
pre-Aut-holomorphic
structure
on
X
top
extends
to
a
unique
Aut-holomorphic
structure
on
X
top
.
Proof.
Assertion
(i)
follows
immediately
from
the
definitions,
by
applying
Propo-
sition
2.2,
(i),
to
sufficiently
small
open
discs
in
X
top
.
Assertion
(ii)
follows
im-
mediately
from
assertion
(i)
by
applying
assertion
(i)
to
automorphisms
of
the
Aut-holomorphic
spaces
determined
by
arbitrary
connected
open
subsets
of
X
top
which
determine
the
same
co-holomorphicization
as
the
identity
automorphism.
54
SHINICHI
MOCHIZUKI
Remark
2.3.1.
Note
that
Corollary
2.3
may
be
thought
of
as
one
sort
of
“com-
plex
analytic
analogue
of
the
Grothendieck
Conjecture”,
that,
although
formulated
somewhat
differently,
contains
[to
a
substantial
extent]
the
same
essential
mathe-
matical
content
as
[Mzk14],
Theorem
1.12
—
cf.
the
similarity
between
the
proofs
of
Proposition
2.2
and
[Mzk14],
Lemma
1.10;
the
application
of
the
p-adic
version
of
Cartan’s
theorem
in
the
proof
of
[Mzk8],
Theorem
1.1
[i.e.,
in
the
proof
of
[Mzk8],
Lemma
1.3].
Remark
2.3.2.
It
follows,
in
particular,
from
Corollary
2.3,
(ii),
that
[in
the
notation
of
Definition
2.1,
(iv)]
the
notion
of
a
co-holomorphicization
X
→
Y
is,
in
fact,
independent
of
the
choice
of
the
local
structures
U
,
V.
Remark
2.3.3.
It
follows
immediately
from
Corollary
2.3,
(i),
that
any
composite
of
morphisms
of
Aut-holomorphic
spaces
is
again
a
morphism
of
Aut-holomorphic
spaces.
Corollary
2.4.
(Holomorphic
Arithmeticity
and
Cores)
Let
X
be
a
hyper-
bolic
Aut-holomorphic
space
of
finite
type
associated
to
a
Riemann
surface
X
[which
is,
in
turn,
determined
by
a
hyperbolic
curve
over
C].
Then
one
may
determine
the
arithmeticity
[in
the
sense
of
[Mzk3],
§2]
of
X
and,
when
X
is
not
arithmetic,
construct
the
Aut-holomorphic
orbispace
[cf.
Remark
2.1.1]
associated
to
the
hyperbolic
core
[cf.
[Mzk3],
Definition
3.1]
of
X,
via
the
following
func-
torial
algorithm,
which
involves
only
the
Aut-holomorphic
space
X
as
input
data:
(a)
Let
U
top
→
X
top
be
any
universal
covering
of
X
top
[i.e.,
a
connected
covering
space
of
the
topological
space
X
top
which
does
not
admit
any
non-
trivial
connected
covering
spaces].
Then
one
may
construct
the
funda-
mental
group
π
1
(X
top
)
as
the
group
of
automorphisms
Aut(U
top
/X
top
)
of
U
top
over
X
top
.
(b)
In
the
notation
of
(a),
by
considering
the
local
structure
on
U
top
consisting
of
connected
open
subsets
of
U
top
that
map
isomorphically
onto
open
subsets
of
X
top
,
one
may
construct
a
natural
pre-Aut-holomorphic
structure
on
U
top
—
hence
also
[cf.
Corollary
2.3,
(ii)]
a
natural
Aut-
holomorphic
structure
on
U
top
—
by
restricting
the
Aut-holomorphic
structure
of
X
on
X
top
;
denote
the
resulting
Aut-holomorphic
space
by
U.
Thus,
we
obtain
a
natural
injection
π
1
(X
top
)
=
Aut(U
top
/X
top
)
→
Aut
0
(U)
⊆
Aut(U)
—
where
we
recall
[cf.
Proposition
2.2,
(ii);
Corollary
2.3,
(i)]
that
Aut(U),
equipped
with
the
compact-open
topology,
is
isomorphic,
as
a
topological
group,
to
GL
2
(R)/R
×
;
we
write
Aut
0
(U)
⊆
Aut(U)
for
the
connected
component
of
the
identity
of
Aut(U).
(c)
In
the
notation
of
(b),
X
is
not
arithmetic
if
and
only
if
the
image
of
π
1
(X
top
)
in
Aut
0
(U)
is
of
finite
index
in
its
commensurator
Π
⊆
TOPICS
IN
ABSOLUTE
ANABELIAN
GEOMETRY
III
55
Aut
0
(U)
in
Aut
0
(U)
[cf.
[Mzk3],
§2,
§3].
If
X
is
not
arithmetic,
then
the
Aut-holomorphic
orbispace
X
→
H
associated
to
the
hyperbolic
core
H
of
X
may
be
constructed
by
form-
ing
the
“orbispace
quotient”
of
U
top
by
Π
and
equipping
this
quotient
with
the
pre-Aut-holomorphic
structure
—
which
[cf.
Corollary
2.3,
(ii)]
deter-
mines
a
unique
Aut-holomorphic
structure
—
determined
by
restrict-
ing
the
Aut-holomorphic
structure
of
U
to
some
suitable
local
structure
as
in
(b).
Finally,
the
asserted
“functoriality”
is
with
respect
to
finite
étale
morphisms
of
Aut-holomorphic
spaces
arising
from
hyperbolic
curves
over
C.
Proof.
The
validity
of
the
algorithm
asserted
in
Corollary
2.4
is
immediate
from
the
constructions
that
appear
in
the
statement
of
this
algorithm
[together
with
the
references
quoted
in
these
constructions].
Remark
2.4.1.
One
verifies
immediately
that
Corollary
2.4
admits
a
natural
extension
to
the
case
where
X
arises
from
a
hyperbolic
orbicurve
over
C
[cf.
Remark
2.1.1].
Remark
2.4.2.
Relative
to
the
analogy
with
the
theory
of
§1
[cf.
Remark
2.7.3
below],
Corollary
2.4
may
be
regarded
as
a
sort
of
holomorphic
analogue
of
results
such
as
[Mzk10],
Theorem
2.4,
concerning
categories
of
finite
étale
localizations
of
hyperbolic
orbicurves.
Next,
we
turn
our
attention
to
re-examining,
from
an
algorithm-based
point
of
view,
the
theory
of
affine
linear
structures
on
Riemann
surfaces
in
the
style
of
[Mzk14],
§2;
[Mzk14],
Appendix.
Following
the
terminology
of
[Mzk14],
Definition
A.3,
(i),
(ii),
we
shall
refer
to
as
“parallelograms”,
“rectangles”,
or
“squares”
the
distinguished
open
subsets
of
C
=
R
+
iR
which
are
of
the
form
suggested
by
these
respective
terms.
Proposition
2.5.
Squares)
Let
(Linear
Structures
via
Parallelograms,
Rectangles,
or
U
⊆
C
=
R
+
iR
be
a
connected
open
subset.
Write
S(U
)
⊆
R(U
)
⊆
P(U
)
for
the
sets
of
pre-compact
squares,
rectangles,
and
parallelograms
in
U
;
let
Q
∈
{S,
R,
P}.
Then
there
exists
a
functorial
algorithm
for
constructing
the
parallel
line
segments,
parallelograms,
orientations,
and
“local
additive
structures”
[in
the
sense
described
below]
of
U
that
involves
only
the
input
data
(U,
Q(U
))
—
i.e.,
consisting
of
the
abstract
set
U
,
equipped
with
the
datum
of
a
56
SHINICHI
MOCHIZUKI
collection
of
distinguished
open
subsets
Q(U
)
[which
clearly
forms
a
basis
for,
hence
determines,
the
topology
of
U
]
—
as
follows:
(a)
Define
a
strict
line
segment
L
of
U
to
be
an
intersection
of
the
form
def
L
=
Q
1
Q
2
—
where
Q
1
,
Q
2
are
the
respective
closures
of
Q
1
,
Q
2
∈
Q(U
);
Q
1
Q
2
=
∅;
L
is
of
infinite
cardinality.
Define
two
strict
line
segments
to
be
strictly
collinear
if
their
intersection
is
of
infinite
cardinality.
Define
a
strict
chain
of
U
to
be
a
finite
ordered
set
of
strict
line
segments
L
1
,
.
.
.
,
L
n
[where
n
≥
2
is
an
integer]
such
that
L
i
,
L
i+1
are
strictly
collinear
for
i
=
1,
.
.
.
,
n
−
1.
Then
one
constructs
the
[closed,
bounded]
line
segments
of
U
by
observing
that
a
line
segment
may
be
character-
ized
as
the
union
of
strict
line
segments
contained
a
strict
chain
of
U
;
an
endpoint
of
a
line
segment
L
is
a
point
of
the
boundary
∂L
of
L
[i.e.,
a
point
whose
complement
in
L
is
connected].
(b)
Define
a
∂Q-parallelogram
of
U
to
be
a
closed
subset
of
U
of
the
form
def
∂Q
=
Q\Q
—
where
Q
∈
Q(U
);
Q
denotes
the
closure
of
Q.
Define
a
side
of
a
parallelogram
Q
∈
Q(U
)
to
be
a
maximal
line
segment
contained
in
the
∂Q-parallelogram
∂Q.
Define
two
line
segments
L,
L
of
U
to
be
strictly
parallel
if
there
exist
non-intersecting
sides
S,
S
of
a
parallel-
ogram
∈
Q(U
)
such
that
S
⊆
L,
S
⊆
L
.
Then
one
constructs
the
pairs
(L,
L
)
of
parallel
line
segments
by
observing
that
L,
L
are
parallel
if
and
only
if
L
is
equivalent
to
L
relative
to
the
equivalence
relation
on
line
segments
generated
by
the
relation
of
inclusion
and
the
relation
of
being
strictly
parallel.
(c)
Define
a
pre-∂-parallelogram
∂P
of
U
to
be
a
union
of
the
mem-
bers
of
a
family
of
four
line
segments
{L
i
}
i∈Z/4Z
of
U
such
that
for
any
two
distinct
points
p
1
,
p
2
∈
∂P
,
there
exists
a
line
segment
L
such
that
∂L
=
{p
1
,
p
2
},
and,
moreover,
for
each
i
∈
Z/4Z,
L
i
and
L
i+2
are
paral-
lel
and
non-intersecting,
and
we
have
an
equality
of
sets
L
i
L
i+1
=
(∂L
i
)
(∂L
i+1
)
of
cardinality
one.
If
∂P
is
a
pre-∂-parallelogram
of
U
,
then
define
the
associated
pre-parallelogram
of
U
to
be
the
union
of
line
segments
L
of
U
such
∂L
⊆
∂P
.
Then
one
constructs
the
parallelograms
∈
P(U
)
of
U
as
the
interiors
of
the
pre-parallelograms
of
U
.
(d)
Let
p
∈
U
.
Define
a
frame
F
=
(S
1
,
S
2
)
of
U
at
p
to
be
an
ordered
pair
of
distinct
intersecting
sides
S
1
,
S
2
of
a
parallelogram
P
∈
P(U
)
such
that
S
1
S
2
=
{p};
in
this
situation,
we
shall
refer
to
any
line
segment
of
U
that
has
infinite
intersection
with
P
as
being
framed
by
F
.
Define
two
frames
F
=
(S
1
,
S
2
),
F
=
(S
1
,
S
2
)
of
U
at
p
to
be
strictly
co-oriented
if
S
1
is
framed
by
F
,
and
S
2
is
framed
by
F
.
Then
one
constructs
the
orientations
of
U
at
p
[of
which
there
are
precisely
2]
by
observing
that
an
orientation
of
U
at
p
may
be
characterized
as
an
equivalence
class
of
frames
of
U
at
p,
relative
to
the
equivalence
relation
on
frames
of
U
at
p
generated
by
the
relation
of
being
strictly
co-oriented.
TOPICS
IN
ABSOLUTE
ANABELIAN
GEOMETRY
III
57
(e)
Let
p
∈
U
.
Then
given
a,
b
∈
U
,
the
sum
a
+
p
b
∈
U
,
relative
to
the
origin
p
—
i.e.,
the
“local
additive
structure”
of
U
at
p
—
may
be
constructed,
whenever
it
is
defined,
in
the
following
fashion:
If
a
=
p,
then
a
+
p
b
=
b;
if
b
=
p,
then
a
+
p
b
=
a.
If
a,
b
=
p,
then
for
P
∈
P(U
)
such
that
P
contains
[distinct]
intersecting
sides
S
a
,
S
b
for
which
S
a
S
b
=
{p},
∂S
a
=
{p,
a},
∂S
b
=
{p,
b},
we
take
a
+
p
b
to
be
the
unique
endpoint
of
a
side
of
P
that
∈
{a,
b,
p}.
[Thus,
“a
+
p
b”
is
defined
for
a,
b
in
some
neighborhood
of
p
in
U
.]
Finally,
the
asserted
“functoriality”
is
with
respect
to
open
immersions
[of
ab-
stract
topological
spaces]
ι
:
U
1
→
U
2
[where
U
1
,
U
2
⊆
C
are
connected
open
subsets]
such
that
ι
maps
Q(U
1
)
into
Q(U
2
).
Proof.
The
validity
of
the
algorithm
asserted
in
Proposition
2.5
is
immediate
from
the
elementary
content
of
the
characterizations
contained
in
the
statement
of
this
algorithm.
Remark
2.5.1.
We
shall
refer
to
a
frame
of
U
at
p
∈
U
as
orthogonal
if
it
arises
from
an
ordered
pair
of
distinct
intersecting
sides
of
a
rectangle
∈
R(U
)
⊆
P(U
).
Proposition
2.6.
(Local
Linear
Holomorphic
Structures
via
Rectangles
or
Squares)
Let
U
,
S,
R,
P,
Q
be
as
in
Proposition
2.5;
suppose
further
that
Q
=
P.
Then
there
exists
a
functorial
algorithm
for
constructing
the
“local
linear
holomorphic
structure”
[in
the
sense
described
below]
of
U
that
involves
only
the
input
data
(U,
Q(U
))
—
i.e.,
consisting
of
the
abstract
topological
space
U
,
equipped
with
the
datum
of
a
collection
of
distinguished
open
subsets
Q(U
)
—
as
follows:
(a)
For
p
∈
U
,
write
A
p
for
the
group
of
automorphisms
of
the
projective
system
of
connected
open
neighborhoods
of
p
in
U
that
are
compatible
with
the
“local
addi-
tive
structures”
of
Proposition
2.5,
(e),
and
preserve
the
orthogonal
frames
and
orientations
[at
p]
of
Proposition
2.5,
(d);
Remark
2.5.1.
Also,
we
equip
A
p
with
the
topology
induced
by
the
topologies
of
the
open
neighborhoods
of
p
that
A
p
acts
on;
note
that
the
“local
additive
struc-
tures”
of
Proposition
2.5,
(e),
determine
an
additive
structure,
hence
also
a
topological
field
structure
on
A
p
{0}.
Then
we
have
a
natural
isomorphism
of
topological
groups
∼
C
×
→
A
p
[induced
by
the
tautological
action
of
C
×
on
C
⊇
U
]
that
is
compatible
with
the
topological
field
structures
on
the
union
of
either
side
with
“{0}”.
In
particular,
one
may
construct
“C
×
at
p”
—
i.e.,
the
“local
linear
holomorphic
structure”
of
U
at
p
—
by
thinking
of
this
“lo-
cal
linear
holomorphic
structure”
as
being
constituted
by
the
topological
58
SHINICHI
MOCHIZUKI
field
A
p
{0},
equipped
with
its
tautological
action
on
the
projective
system
of
open
neighborhoods
of
p.
(b)
For
p,
p
∈
U
,
one
constructs
a
natural
isomorphism
of
topological
groups
∼
A
p
→
A
p
that
is
compatible
with
the
topological
field
structures
on
either
side
as
follows:
If
p
is
sufficiently
close
to
p,
then
the
“local
additive
struc-
tures”
of
Proposition
2.5,
(e),
determine
homeomorphisms
[by
“trans-
lation”,
i.e.,
“addition”]
from
sufficiently
small
neighborhoods
of
p
onto
sufficiently
small
neighborhoods
of
p
;
these
homeomorphisms
thus
induce
∼
the
desired
isomorphism
A
p
→
A
p
.
Now,
by
joining
an
arbitrary
p
to
p
via
a
chain
of
“sufficiently
small
open
neighborhoods”
and
composing
the
resulting
isomorphisms
of
“local
linear
holomorphic
structures”,
one
ob-
∼
tains
the
desired
isomorphism
A
p
→
A
p
for
arbitrary
p,
p
∈
U
.
Finally,
this
isomorphism
is
independent
of
the
choice
of
a
chain
of
“sufficiently
small
open
neighborhoods”
used
in
its
construction.
Finally,
the
asserted
“functoriality”
is
to
be
understood
in
the
same
sense
as
in
Proposition
2.5.
Proof.
The
validity
of
the
algorithm
asserted
in
Proposition
2.6
is
immediate
from
the
elementary
content
of
the
characterizations
contained
in
the
statement
of
this
algorithm.
Remark
2.6.1.
Thus,
the
algorithms
of
Propositions
2.5,
2.6
may
be
regarded
as
superseding
the
techniques
applied
in
the
proof
of
[Mzk14],
Proposition
A.4.
Moreover,
just
as
the
theory
of
[Mzk14],
Appendix,
was
applied
in
[Mzk14],
§2,
one
may
apply
the
algorithms
of
Propositions
2.5,
2.6
to
give
algorithms
for
recon-
structing
the
local
linear
and
orthogonal
structures
on
a
Riemann
surface
equipped
with
a
nonzero
square
differential
from
the
various
categories
which
are
the
topic
of
[Mzk14],
Theorem
2.3.
We
leave
the
routine
details
to
the
interested
reader.
Corollary
2.7.
(Local
Linear
Holomorphic
Structures
via
Holomorphic
Elliptic
Cuspidalization)
Let
X
be
an
elliptically
admissible
Aut-holomorphic
orbispace
[cf.
Remark
2.1.1]
associated
to
a
Riemann
orbisurface
X.
Then
there
exists
a
functorial
algorithm
for
constructing
the
“local
linear
holo-
morphic
structure”
[cf.
Proposition
2.6]
on
X
top
that
involves
only
the
Aut-
holomorphic
space
X
as
input
data,
as
follows:
(a)
By
the
definition
of
“elliptically
admissible”,
we
may
apply
Corollary
2.4,
(c),
to
construct
the
[Aut-holomorphic
orbispace
associated
to
the]
semi-elliptic
hyperbolic
core
X
→
H
of
X
[i.e.,
X],
together
with
the
unique
[cf.
[Mzk21],
Remark
3.1.1]
double
covering
E
→
H
by
an
Aut-
holomorphic
space
[i.e.,
the
covering
determined
by
the
unique
torsion-
free
subgroup
of
index
two
of
the
group
Π
of
Corollary
2.4,
(c)].
[Thus,
TOPICS
IN
ABSOLUTE
ANABELIAN
GEOMETRY
III
E
is
the
Aut-holomorphic
space
associated
to
a
once-punctured
elliptic
curve.]
(b)
By
considering
“elliptic
cuspidalization
diagrams”
as
in
[Mzk21],
Example
3.2
[cf.
also
the
equivalence
of
Corollary
2.3,
(i)]
E
←
U
→
E
—
where
U
→
E
is
an
abelian
finite
étale
covering
[which
necessarily
extends
to
a
covering
of
the
one-point
compactification
of
E
top
];
E
top
←
U
top
is
an
open
immersion
whose
image
is
the
complement
of
a
finite
subset
of
E
top
;
E
←
U,
U
→
E
are
co-holomorphic
—
one
may
construct
the
torsion
points
of
[the
elliptic
curve
determined
by]
E
as
the
points
in
the
complement
of
the
image
of
such
morphisms
U
→
E,
together
with
the
group
structure
on
these
torsion
points
[which
is
induced
by
the
group
structure
of
the
Galois
group
Gal(U/E)].
(c)
Since
the
torsion
points
of
(b)
are
dense
in
E
top
,
one
may
construct
the
group
structure
on
[the
one-point
compactification
of
]
E
top
[that
arises
from
the
elliptic
curve
determined
by
E]
as
the
unique
topolog-
ical
group
structure
that
extends
the
group
structure
on
the
torsion
points
of
(b).
This
group
structure
determines
“local
additive
struc-
tures”
[cf.
Proposition
2.5,
(e)]
at
the
various
points
of
E
top
.
Moreover,
by
considering
one-parameter
subgroups
of
these
local
additive
group
struc-
tures,
one
constructs
the
line
segments
[cf.
Proposition
2.5,
(a)]
of
E
top
;
by
considering
translations
of
line
segments,
relative
to
these
local
additive
group
structures,
one
constructs
the
pairs
of
parallel
line
segments
[cf.
Proposition
2.5,
(b)]
of
E
top
,
hence
also
the
parallelograms,
frames,
and
orientations
[cf.
Proposition
2.5,
(c),
(d)]
of
E
top
.
(d)
Let
V
be
the
Aut-holomorphic
space
determined
by
a
parallelogram
V
top
⊆
E
top
[cf.
(c)].
Then
the
one-parameter
subgroups
of
the
[topological]
group
A
V
(V
top
)
[
∼
=
SL
2
(R)/{±1}
—
cf.
Proposition
2.2,
(ii);
Corollary
2.3,
(i);
the
Riemann
mapping
theorem
of
elementary
complex
analysis]
are
precisely
the
closed
connected
subgroups
for
which
the
complement
of
some
connected
open
neighborhood
of
the
identity
element
fails
to
be
connected.
If
S
is
a
one-parameter
subgroup
of
A
V
(V
top
),
p
∈
V
top
,
and
L
is
a
line
segment
one
of
whose
endpoints
is
equal
to
p,
then
L
is
tangent
to
S
·
p
at
p
if
and
only
if
any
pairs
of
sequences
of
points
of
L\{p},
(S
·
p)\{p},
converge
to
the
same
element
of
the
quotient
space
V
top
\{p}
P(V,
p)
determined
by
identifying
positive
real
multiples
of
elements
of
V
top
\{p},
relative
to
the
local
additive
structure
at
p.
In
particular,
one
may
con-
struct
the
orthogonal
frames
of
E
top
as
the
frames
consisting
of
pairs
of
line
segments
L
1
,
L
2
emanating
from
a
point
p
∈
E
top
that
are
tan-
gent,
respectively,
to
orbits
S
1
·
p,
S
2
·
p
of
one-parameter
subgroups
S
1
,
S
2
⊆
A
V
(V
top
)
such
that
S
2
is
obtained
from
S
1
by
conjugating
S
1
by
an
element
of
order
four
[i.e.,
“±i”]
of
a
compact
one-parameter
subgroup
[i.e.,
a
“one-dimensional
torus”]
of
A
V
(V
top
)
that
fixes
p.
59
60
SHINICHI
MOCHIZUKI
(e)
For
p
∈
E
top
,
write
A
p
for
the
group
of
automorphisms
of
the
projective
system
of
connected
open
neighborhoods
of
p
in
E
top
that
are
compatible
with
the
“local
ad-
ditive
structures”
of
(c)
and
preserve
the
orthogonal
frames
and
orientations
[at
p]
of
(c),
(d)
[cf.
Proposition
2.6,
(a)].
Then
just
as
in
Proposition
2.6,
(a),
we
obtain
topological
field
structures
on
∼
A
p
{0},
together
with
compatible
isomorphisms
A
p
→
A
p
,
for
p
∈
E
top
.
This
system
of
“A
p
’s”
may
be
thought
of
as
a
system
of
“local
linear
holomorphic
structures”
on
E
top
or
X
top
.
Finally,
the
asserted
“functoriality”
is
with
respect
to
finite
étale
morphisms
of
Aut-holomorphic
orbispaces
arising
from
hyperbolic
orbicurves
over
C.
Proof.
The
validity
of
the
algorithm
asserted
in
Corollary
2.7
is
immediate
from
the
constructions
that
appear
in
the
statement
of
this
algorithm
[together
with
the
references
quoted
in
these
constructions].
Remark
2.7.1.
It
is
by
no
means
the
intention
of
the
author
to
assert
that
the
technique
applied
in
Corollary
2.7,
(b),
(c),
to
recover
the
“local
additive
structure”
via
elliptic
cuspidalization
is
the
unique
way
to
construct
this
local
additive
struc-
ture.
Indeed,
perhaps
the
most
direct
approach
to
the
problem
of
constructing
the
local
additive
structure
is
to
compactify
the
given
once-punctured
elliptic
curve
and
then
to
consider
the
group
structure
of
the
[connected
component
of
the
identity
of
the]
holomorphic
automorphism
group
of
the
resulting
elliptic
curve.
By
com-
parison
to
this
direct
approach,
however,
the
technique
of
elliptic
cuspidalization
has
the
virtue
of
being
compatible
with
the
“hyperbolic
structure”
of
the
hyperbolic
orbicurves
involved.
In
particular,
it
is
compatible
with
the
various
“hyperbolic
fun-
damental
groups”
of
these
orbicurves.
This
sort
of
compatibility
with
fundamental
groups
plays
an
essential
role
in
the
nonarchimedean
theory
[cf.,
e.g.,
the
theory
of
[Mzk18],
§1,
§2].
On
the
other
hand,
the
“direct
approach”
described
above
is
not
entirely
unrelated
to
the
approach
via
elliptic
cuspidalization
in
the
sense
that,
if
one
thinks
of
the
torsion
points
in
the
latter
approach
as
playing
an
analogous
role
to
the
role
played
by
the
“entire
compactified
elliptic
curve”
in
the
former
approach,
then
the
latter
approach
may
be
thought
of
as
a
sort
of
discretization
via
torsion
points
—
cf.
the
point
of
view
of
Hodge-Arakelov
theory,
as
discussed
in
[Mzk6],
[Mzk7]
—
of
the
former
approach.
Here,
we
note
that
the
density
of
torsion
points
in
the
archimedean
theory
of
the
elliptic
cuspidalization
is
reminiscent
of
the
density
of
NF-points
in
the
nonarchimedean
theory
of
the
Belyi
cuspidalization
[cf.
§1].
Remark
2.7.2.
In
light
of
the
role
played
by
the
technique
of
elliptic
cuspidal-
ization
both
in
Corollary
2.7
and
in
the
theory
of
[Mzk18],
§1,
§2,
it
is
of
interest
to
compare
these
two
theories.
From
an
archimedean
point
of
view,
the
theory
of
[Mzk18]
may
be
roughly
summarized
as
follows:
One
begins
with
the
uniformization
∼
G
G/q
Z
→
E
TOPICS
IN
ABSOLUTE
ANABELIAN
GEOMETRY
III
61
of
an
elliptic
curve
E
over
C
by
a
copy
G
of
C
×
.
Here,
the
“q-parameter”
of
E
may
be
thought
of
as
being
an
element
def
q
∈
H
=
G
⊗
Gal(G/E)
[where
we
recall
that
Gal(G/E)
∼
=
Z].
Then
one
thinks
of
the
theta
function
associated
to
E
as
a
function
Θ
:
G
→
H
[i.e.,
a
function
defined
on
G
with
values
in
H].
From
this
point
of
view,
the
various
types
of
rigidity
considered
in
the
theory
of
[Mzk18]
may
be
understood
in
the
following
fashion:
(a)
Cyclotomic
rigidity
corresponds
to
the
portion
of
the
tautological
iso-
∼
morphism
H
→
G
⊗
Gal(G/E)
involving
the
maximal
compact
subgroups,
def
i.e.,
the
copies
of
S
1
=
{z
∈
C
×
|
|z|
=
1}
⊆
C
×
.
(b)
Discrete
rigidity
corresponds
to
the
portion
of
the
tautological
isomor-
∼
phism
H
→
G⊗Gal(G/E)
involving
the
quotients
by
the
maximal
compact
def
subgroups,
i.e.,
the
copies
of
R
>0
=
{z
∈
R
|
z
>
0}
∼
=
C
×
/S
1
.
(c)
Constant
rigidity
corresponds
to
considering
the
normalization
of
√
Θ
given
by
taking
the
values
of
Θ
at
the
points
of
G
corresponding
to
±
−1
to
be
±1.
In
particular,
the
“canonical
copy
of
C
×
”
that
arises
from
(a),
(b)
—
i.e.,
H
—
is
related
to
the
“copies
of
C
×
”
that
occur
as
the
“A
p
”
of
Corollary
2.7,
(e),
in
the
following
way:
A
p
is
given
by
the
linear
holomorphic
automorphisms
of
the
tangent
space
to
a
point
of
H.
That
is
to
say,
roughly
speaking,
A
p
(
∼
=
C
×
)
is
related
to
H
(
∼
=
C
×
)
by
the
operation
of
“taking
the
logarithm”,
followed
by
the
operation
of
“taking
Aut(−)”
[of
the
resulting
linearization].
Remark
2.7.3.
It
is
interesting
to
note
that
just
as
the
absolute
Galois
group
G
k
of
an
MLF
k
may
be
regarded
as
a
two-dimensional
object
with
one
rigid
and
one
non-rigid
dimension
[cf.
Remark
1.9.4],
the
topological
group
C
×
is
also
a
two-dimensional
object
with
one
rigid
dimension
—
i.e.,
S
1
=
{z
∈
C
×
|
|z|
=
1}
⊆
C
×
def
[a
topological
group
whose
automorphism
group
is
of
order
2]
—
and
one
non-rigid
dimension
—
i.e.,
def
R
>0
=
{z
∈
R
|
z
>
0}
⊆
C
×
[a
topological
group
that
is
isomorphic
to
R,
hence
has
automorphism
group
given
by
R
×
—
i.e.,
a
“continuous
family
of
dilations”].
Moreover,
just
as,
in
the
context
of
Theorem
1.9,
Corollary
1.10,
considering
G
k
equipped
with
its
outer
action
on
Δ
X
has
the
effect
of
rendering
both
dimensions
of
G
k
rigid
[cf.
Remark
1.9.4],
considering
“C
×
”
as
arising,
in
the
fashion
discussed
in
Corollary
2.7,
from
a
certain
Aut-holomorphic
orbispace
has
the
effect
of
rigidifying
both
dimensions
of
C
×
.
We
refer
to
Remark
2.7.4
below
for
more
on
this
analogy
between
the
(i)
outer
action
of
G
k
on
Δ
X
62
SHINICHI
MOCHIZUKI
and
the
notion
of
an
(ii)
Aut-holomorphic
orbispace
associated
to
a
hyperbolic
orbicurve.
Finally,
we
observe
that
from
the
point
of
view
of
the
problem
of
finding
an
algorithm
to
construct
the
base
field
of
a
hyperbolic
orbicurve
from
(i),
(ii),
one
may
think
of
Theorem
1.9
and
Corollaries
1.10,
2.7
as
furnishing
solutions
to
various
versions
of
this
problem.
Remark
2.7.4.
The
usual
definition
of
a
“holomorphic
structure”
on
a
Riemann
surface
is
via
local
comparison
to
some
fixed
model
of
the
topological
field
C.
The
local
homeomorphisms
that
enable
this
comparison
are
related
to
one
another
by
homeomorphisms
of
open
neighborhoods
of
C
that
are
holomorphic.
On
the
other
hand,
this
definition
does
not
yield
any
absolute
description
—
i.e.,
a
description
that
depends
on
mathematical
structures
that
do
not
involve
explicit
use
of
models
—
of
what
precisely
is
meant
by
the
notion
of
a
“holomorphic
structure”.
Instead,
it
relies
on
relating/comparing
the
given
manifold
to
the
fixed
model
of
C
—
an
approach
that
is
“model-explicit”.
By
contrast,
the
notion
of
a
topological
space
[i.e.,
consisting
of
the
datum
of
a
collection
of
subsets
that
are
to
be
regarded
as
“open”]
is
absolute,
or
“model-implicit”.
In
a
similar
vein,
the
approach
to
quasi-
conformal
or
conformal
structures
via
the
datum
of
a
collection
of
parallelograms,
rectangles,
or
squares
[cf.
Propositions
2.5,
2.6;
Remark
2.6.1;
the
theory
of
[Mzk14]]
is
“model-implicit”.
The
approach
to
“holomorphic
structures”
on
a
Riemann
sur-
face
via
the
classical
notion
of
a
“conformal
structure”
[i.e.,
the
datum
of
various
orthogonal
pairs
of
tangent
vectors]
is,
so
to
speak,
“relatively
model-implicit”,
i.e.,
“model-implicit”
modulo
the
fact
that
it
depends
on
the
“model-explicit”
definition
of
the
notion
of
a
differential
manifold
—
which
may
be
thought
of
as
a
sort
of
“lo-
cal
linear
structure”
that
is
given
by
local
comparison
to
the
local
linear
structure
of
Euclidean
space.
From
this
point
of
view:
The
notions
of
an
“outer
action
of
G
k
on
Δ
X
”
and
an
“Aut-holomorphic
orbispace”
[cf.
Remark
2.7.3,
(i),
(ii)]
have
the
virtue
of
being
“model-
implicit”
—
i.e.,
they
do
not
depend
on
any
sort
of
[local]
comparison
to
some
fixed
reference
model.
In
this
context,
it
is
interesting
to
note
that
all
of
the
examples
given
so
far
of
“model-implicit”
definitions
depend
on
data
consisting
either
of
subsets
[e.g.,
open
subsets
of
a
topological
space;
parallelograms,
rectangles,
or
squares
on
a
Riemann
surface]
or
endomorphisms
[e.g.,
the
automorphisms
that
appear
in
a
Galois
cat-
egory;
the
automorphisms
that
appear
in
an
Aut-holomorphic
structure].
[Here,
in
passing,
we
note
that
the
appearance
of
“endomorphisms”
in
the
present
discussion
is
reminiscent
of
the
discussion
of
“hidden
endomorphisms”
in
the
Introduction
to
[Mzk21].]
Also,
we
observe
that
this
dichotomy
between
model-explicit
and
model-
implicit
definitions
is
strongly
reminiscent
of
the
distinction
between
bi-anabelian
and
mono-anabelian
geometry
discussed
in
Remark
1.9.8.
TOPICS
IN
ABSOLUTE
ANABELIAN
GEOMETRY
III
63
Finally,
we
relate
the
archimedean
theory
of
the
present
§2
to
the
Galois-
theoretic
theory
of
§1,
in
the
case
of
number
fields,
via
a
sort
of
archimedean
analogue
of
Corollary
1.10.
Corollary
2.8.
(Galois-theoretic
Reconstruction
of
Aut-holomorphic
Spaces)
Let
X,
k
⊆
k
⊇
k
NF
,
and
1
→
Δ
X
→
Π
X
→
G
k
→
1
be
as
in
Theorem
1.9;
suppose
further
that
k
is
a
number
field
[so
k
NF
=
k],
and
[for
simplic-
ity
—
cf.
Remark
2.8.2
below]
that
X
is
a
curve.
Then
one
may
think
of
each
×
archimedean
prime
of
the
field
k
NF
{0}
(
∼
=
k
NF
)
constructed
in
Theorem
1.9,
×
(e),
as
a
topology
on
k
NF
{0}
satisfying
certain
properties.
Moreover,
for
each
such
archimedean
prime
v,
there
exists
a
functorial
“group-theoretic”
al-
gorithm
for
reconstructing
the
Aut-holomorphic
space
X
v
associated
to
def
X
v
=
X
×
k
k
v
×
[where
we
write
k
v
for
the
completion
of
k
NF
{0}
at
v]
from
the
topological
group
Π
X
;
this
algorithm
consists
of
the
following
steps:
(a)
Define
a
Cauchy
sequence
{x
j
}
j∈N
of
NF-points
[of
X
v
]
to
be
a
se-
quence
of
NF-points
x
j
[i.e.,
conjugacy
classes
of
decomposition
groups
of
NF-points
in
Π
X
—
cf.
Theorem
1.9,
(a)]
such
that
there
exists
a
finite
set
of
NF-points
S
—
which
we
shall
refer
to
as
a
conductor
for
the
Cauchy
sequence
—
satisfying
the
following
two
conditions:
(i)
x
j
∈
S
for
all
but
finitely
many
j
∈
N;
(ii)
for
every
NF-rational
function
f
on
X
k
as
in
Theorem
1.9,
(d),
whose
divisor
of
poles
avoids
S,
the
sequence
of
[non-infinite,
for
all
but
finitely
many
j
—
cf.
(i)]
values
{f
(x
j
)
∈
k
v
}
j∈N
forms
a
Cauchy
sequence
[in
the
usual
sense]
of
k
v
.
Two
Cauchy
se-
quences
{x
j
}
j∈N
,
{y
j
}
j∈N
of
NF-points
which
admit
a
common
conductor
S
will
be
called
equivalent
if
for
every
NF-rational
function
f
on
X
k
as
in
Theorem
1.9,
(d),
whose
divisor
of
poles
avoids
S,
the
sequences
of
[non-infinite,
for
all
but
finitely
many
j]
values
{f
(x
j
)}
j∈N
,
{f
(y
j
)}
j∈N
form
Cauchy
sequences
in
k
v
that
converge
to
the
same
element
of
k
v
.
For
U
⊆
k
v
an
open
subset
and
f
an
NF-rational
function
on
X
k
as
in
Theorem
1.9,
(d),
we
obtain
a
set
N
(U,
f
)
of
Cauchy
sequences
of
NF-
points
by
considering
the
Cauchy
sequences
of
NF-points
{x
j
}
j∈N
such
that
f
(x
j
)
[is
finite
and]
∈
U
,
for
all
j
∈
N.
Then
one
constructs
the
topological
space
X
top
=
X
v
(k
v
)
as
the
set
of
equivalence
classes
of
Cauchy
sequences
of
NF-points,
equipped
with
the
topology
defined
by
the
sets
“N
(U,
f
)”.
(b)
Let
U
X
⊆
X
top
,
U
v
⊆
k
v
be
connected
open
subsets
and
f
a
NF-
rational
function
on
X
k
as
in
Theorem
1.9,
(d),
such
that
the
func-
tion
defined
by
f
on
U
X
[i.e.,
by
taking
limits
of
Cauchy
sequences
of
∼
values
in
k
v
—
cf.
(a)]
determines
a
homeomorphism
f
U
:
U
X
→
U
v
.
∼
Write
Aut
hol
(U
v
)
for
the
group
of
self-homeomorphisms
U
v
→
U
v
(⊆
k
v
),
which,
relative
to
the
topological
field
structure
of
k
v
,
can
locally
[on
64
SHINICHI
MOCHIZUKI
U
v
]
be
expressed
as
a
convergent
power
series
with
coefficients
in
k
v
;
A
X
(U
X
)
=
f
U
−1
◦
Aut
hol
(U
v
)
◦
f
U
⊆
Aut(U
X
).
Then
one
constructs
the
Aut-holomorphic
structure
A
X
on
X
top
as
the
unique
[cf.
Corollary
2.3,
(ii)]
Aut-holomorphic
structure
that
extends
the
pre-Aut-holomorphic
structure
determined
by
the
groups
“A
X
(U
X
)”;
we
take
X
v
to
be
the
Aut-
holomorphic
space
determined
by
the
objects
(X
top
,
A
X
).
def
Finally,
the
asserted
“functoriality”
is
with
respect
to
arbitrary
open
injective
homomorphisms
of
profinite
groups
[i.e.,
of
“Π
X
”]
that
are
compatible
with
the
respective
choices
of
archimedean
valuations
[i.e.,
“v”].
Proof.
The
validity
of
the
algorithm
asserted
in
Corollary
2.8
is
immediate
from
the
constructions
that
appear
in
the
statement
of
this
algorithm
[together
with
the
references
quoted
in
these
constructions].
Remark
2.8.1.
One
verifies
immediately
that
the
isomorphism
class
of
the
pair
(1
→
Δ
X
→
Π
X
→
G
k
→
1,
v)
depends
only
on
the
restriction
of
v
to
the
subfield
×
k
×
{0}
⊆
k
NF
{0}.
Remark
2.8.2.
One
verifies
immediately
that
Corollary
2.8
[as
well
as
Corollary
2.9
below]
may
be
extended
to
the
case
where
X
is
a
hyperbolic
orbicurve
that
is
not
necessarily
a
curve
[so
X
v
will
be
an
Aut-holomorphic
orbispace].
Remark
2.8.3.
One
verifies
immediately
that
any
elliptically
admissible
hyper-
bolic
orbicurve
defined
over
a
number
field
is
of
strictly
Belyi
type.
In
particular,
if
one
is
given
an
elliptically
admissible
hyperbolic
orbicurve
X
that
is
defined
over
a
number
field
k,
then
it
makes
sense
to
apply
Corollary
2.7
to
the
Aut-holomorphic
[orbi]spaces
constructed
in
Corollary
2.8.
This
compatibility
between
Corollaries
2.7,
2.8
[cf.
also
Corollary
2.9
below]
is
one
reason
why
it
is
of
interest
to
construct
the
local
additive
structures
as
in
Corollary
2.7,
(c),
directly
from
the
Aut-holomorphic
structure
as
opposed
to
via
the
“parallelogram-theoretic”
approach
of
Proposition
2.5,
2.6
[cf.
also
Remark
2.6.1],
which
is
more
suited
to
“strictly
archimedean
situations”
—
i.e.,
situations
in
which
one
is
not
concerned
with
regarding
Aut-
holomorphic
orbispaces
as
arising
from
hyperbolic
orbicurves
over
number
fields.
Corollary
2.9.
(Global-Archimedean
Elliptically
Admissible
Compat-
ibility)
In
the
notation
of
Corollary
2.8,
suppose
further
that
X
is
elliptically
admissible;
take
the
Aut-holomorphic
space
X
of
Corollary
2.7
to
be
the
Aut-
holomorphic
space
determined
by
the
objects
(X
top
,
A
X
)
constructed
in
Corollary
2.8.
Then
one
may
construct,
in
a
functorially
algorithmic
fashion,
an
iso-
morphism
between
the
topological
field
k
v
of
Corollary
2.8
and
the
topological
fields
“A
p
{0}”
of
Corollary
2.7,
(e),
in
the
following
way:
(a)
Let
x
∈
X
v
(k
v
)
be
an
NF-point.
The
local
additive
structures
on
E
top
[cf.
Corollary
2.7,
(c)]
determine
local
additive
structures
on
X
top
;
let
v
be
an
element
of
a
sufficiently
small
neighborhood
U
X
⊆
X
top
of
x
in
X
top
TOPICS
IN
ABSOLUTE
ANABELIAN
GEOMETRY
III
65
that
admits
such
a
local
additive
structure.
Then
for
each
NF-rational
function
f
that
vanishes
at
x,
the
assignment
(v
,
f
)
→
lim
n
·
f
(
n→∞
1
·
x
v
)
∈
k
v
n
[where
“·
x
”
is
the
operation
arising
from
the
local
additive
structure
at
x]
depends
only
on
the
image
df
|
x
∈
ω
x
of
f
in
the
Zariski
cotangent
space
ω
x
to
X
v
at
x
and,
moreover,
determines
a
topological
embedding
ι
U
X
,x
:
U
X
→
Hom
k
v
(ω
x
,
k
v
)
that
is
compatible
with
the
“local
additive
structures”
of
the
domain
and
codomain.
(b)
By
letting
the
neighborhoods
U
X
of
a
fixed
NF-point
x
vary,
the
resulting
ι
U
X
,x
determine
an
isomorphism
of
topological
fields
A
x
∼
{0}
→
k
v
via
the
condition
of
compatibility
[with
respect
to
the
ι
U
X
,x
]
with
the
nat-
ural
actions
of
A
x
,
k
v
,
respectively,
on
the
domain
and
codomain
of
ι
U
X
,x
.
Moreover,
as
x
varies,
these
isomorphisms
are
compatible
with
the
iso-
∼
morphisms
A
x
1
{0}
→
A
x
2
{0}
[where
x
1
,
x
2
∈
X(k
v
)
are
NF-points]
of
Corollary
2.7,
(e).
Finally,
the
asserted
“functoriality”
is
to
be
understood
in
the
sense
described
in
Corollary
2.8.
Proof.
The
validity
of
the
algorithm
asserted
in
Corollary
2.9
is
immediate
from
the
constructions
that
appear
in
the
statement
of
this
algorithm
[together
with
the
references
quoted
in
these
constructions].
66
SHINICHI
MOCHIZUKI
Section
3:
Nonarchimedean
Log-Frobenius
Compatibility
In
the
present
§3,
we
give
an
interpretation
of
the
nonarchimedean
local
portion
of
the
theory
of
§1
in
terms
of
a
certain
compatibility
with
the
“log-Frobenius
func-
tor”
[in
essence,
a
version
of
the
usual
“logarithm”
at
the
various
nonarchimedean
primes
of
a
number
field].
In
order
to
express
this
compatibility,
certain
abstract
category-theoretic
ideas
—
which
center
around
the
notions
of
observables,
telecores,
and
cores
—
are
introduced
[cf.
Definition
3.5].
These
notions
allow
one
to
express
the
log-Frobenius
compatibility
of
the
mono-anabelian
construction
algorithms
of
§1
[cf.
Corollary
3.6],
as
well
as
the
failure
of
log-Frobenius
compatibility
that
occurs
if
one
attempts
to
take
a
“bi-anabelian”
approach
to
the
situation
[cf.
Corollary
3.7].
Definition
3.1.
def
(i)
Let
k
be
an
MLF,
k
an
algebraic
closure
of
k,
G
k
=
Gal(k/k).
Write
O
k
⊆
k
for
the
ring
of
integers
of
k,
O
k
×
⊆
O
k
for
the
group
of
units
of
O
k
,
and
O
k
⊆
O
k
for
the
multiplicative
monoid
of
nonzero
elements
[cf.
[Mzk17],
Example
1.1,
(i)];
we
shall
use
similar
notation
for
other
subfields
of
k.
Let
Π
k
be
a
topological
group,
equipped
with
a
continuous
surjection
k
:
Π
k
G
k
.
Note
that
the
[p-adic,
if
k
is
of
residue
characteristic
p]
logarithm
determines
a
Π
k
-equivariant
isomorphism
∼
log
k
:
k
∼
=
(O
×
)
pf
→
k
def
k
[where
“pf”
denotes
the
perfection
[cf.,
e.g.,
[Mzk16],
§0];
the
Π
k
-action
is
the
action
obtained
by
composing
with
k
]
of
the
topological
group
k
∼
onto
the
additive
topological
group
k.
Next,
let
us
refer
to
an
abelian
monoid
[e.g.,
an
abelian
group]
whose
subgroup
of
torsion
elements
is
[abstractly]
isomorphic
to
Q/Z
as
torsion-
cyclotomic;
let
T
be
one
of
the
following
categories
[cf.
§0
for
more
on
the
prefix
“ind-”]:
·
TF:
ind-topological
fields
and
homomorphisms
of
ind-topological
fields;
·
TCG:
torsion-cyclotomic
ind-compact
abelian
topological
groups
and
ho-
momorphisms
of
ind-topological
groups;
·
TLG:
torsion-cyclotomic
ind-locally
compact
abelian
topological
groups
and
homomorphisms
of
ind-topological
groups;
·
TM:
torsion-cyclotomic
ind-topological
abelian
monoids
and
homomor-
phisms
of
ind-topological
monoids;
·
TS:
ind-locally
compact
topological
spaces
and
morphisms
of
ind-topological
spaces;
·
TS:
ind-locally
compact
abelian
topological
groups
and
homomorphisms
of
ind-topological
groups
[so
we
have
a
natural
full
embedding
TLG
→
TS].
If
T
is
equal
to
TF
(respectively,
TCG;
TLG;
TM;
TS;
TS),
then
let
M
k
∈
Ob(T)
be
the
object
determined
by
k
(respectively,
the
object
determined
by
O
×
;
k
TOPICS
IN
ABSOLUTE
ANABELIAN
GEOMETRY
III
67
×
the
object
determined
by
k
;
the
object
determined
by
O
;
any
object
of
TS
k
equipped
with
a
faithful
continuous
G
k
-action;
any
object
of
TS
equipped
with
a
faithful
continuous
G
k
-action).
We
shall
refer
to
as
a
model
MLF-Galois
T-pair
any
collection
of
data
(a),
(b),
(c)
of
the
following
form:
(a)
the
topological
group
Π
k
,
(b)
the
object
M
k
∈
Ob(T),
(c)
the
action
of
Π
k
on
M
k
[so
the
quotient
Π
k
G
k
may
be
recovered
as
the
image
of
the
homomorphism
Π
k
→
Aut(M
k
)
arising
from
the
action
of
(c)];
we
shall
often
use
the
abbreviated
notation
(Π
k
M
k
)
for
this
collection
of
data
(a),
(b),
(c).
(ii)
We
shall
refer
to
any
collection
of
data
(Π
M
)
consisting
of
a
topological
group
Π,
an
object
M
∈
Ob(T),
and
a
continuous
action
of
Π
on
M
as
an
MLF-
Galois
T-pair
if,
for
some
model
MLF-Galois
T-pair
(Π
k
M
k
)
[where
the
notation
∼
is
as
in
(i)],
there
exist
an
isomorphism
of
topological
groups
Π
k
→
Π
and
an
∼
isomorphism
of
objects
M
k
→
M
of
T
that
are
compatible
with
the
respective
actions
of
Π
k
,
Π
on
M
k
,
M
;
in
this
situation,
we
shall
refer
to
Π
as
the
Galois
group,
to
the
surjection
Π
G
determined
by
the
action
of
Π
on
M
[cf.
(i)]
as
the
Galois
augmentation,
to
G
as
the
arithmetic
Galois
group,
and
to
M
as
the
arithmetic
data
of
the
MLF-Galois
T-pair
(Π
M
);
if,
in
this
situation,
the
surjection
Π
k
G
k
arises
from
the
étale
fundamental
group
of
an
arbitrary
hyperbolic
orbicurve
(respectively,
a
hyperbolic
orbicurve
of
strictly
Belyi
type)
over
k,
then
we
shall
refer
to
the
MLF-Galois
T-pair
(Π
M
)
as
being
of
hyperbolic
orbicurve
type
(respectively,
of
strictly
Belyi
type);
if,
in
this
situation,
the
surjection
Π
k
G
k
is
an
isomorphism,
then
we
shall
refer
to
the
MLF-Galois
T-pair
(Π
M
)
as
being
of
mono-analytic
type
[cf.
Remark
5.6.1
below
for
more
on
this
terminology].
A
morphism
of
MLF-Galois
T-pairs
φ
:
(Π
1
M
1
)
→
(Π
2
M
2
)
consists
of
a
morphism
of
objects
φ
M
:
M
1
→
M
2
of
T,
together
with
a
compatible
[relative
to
the
respective
actions
of
Π
1
,
Π
2
on
M
1
,
M
2
]
continuous
homomorphism
of
topological
groups
φ
Π
:
Π
1
→
Π
2
that
induces
an
open
injective
homomor-
phism
between
the
respective
arithmetic
Galois
groups;
if,
in
this
situation,
φ
M
(respectively,
φ
Π
)
is
an
isomorphism,
then
we
shall
refer
to
φ
as
a
T-isomorphism
(respectively,
Galois-isomorphism).
(iii)
Write
C
T
MLF
for
the
category
whose
objects
are
the
MLF-Galois
T-pairs
and
whose
morphisms
are
the
morphisms
of
MLF-Galois
T-pairs.
Also,
we
shall
use
the
same
notation,
except
with
“C”
replaced
by
C
(respectively,
C;
C)
to
denote
the
various
subcategories
determined
by
the
T-isomorphisms
(respec-
tively,
Galois-isomorphisms;
isomorphisms);
we
shall
use
the
same
notation,
with
“MLF”
replaced
by
MLF-hyp
(respectively,
MLF-sB;
MLF)
68
SHINICHI
MOCHIZUKI
to
denote
the
various
full
subcategories
determined
by
the
objects
of
hyperbolic
orbicurve
type
(respectively,
of
strictly
Belyi
type;
of
mono-analytic
type).
Since
×
[in
the
notation
of
(i)]
the
formation
of
O
(respectively,
k
;
O
×
;
O
×
)
from
k
k
×
k
k
(respectively,
O
;
O
;
k
)
is
clearly
intrinsically
defined
[i.e.,
depends
only
on
the
k
k
“input
data
of
an
object
of
T”],
we
thus
obtain
natural
functors
MLF
MLF
→
C
TM
;
C
TF
MLF
MLF
C
TM
→
C
TLG
;
MLF
MLF
C
TM
→
C
TCG
;
MLF
MLF
C
TLG
→
C
TCG
—
i.e.,
by
taking
the
multiplicative
group
of
nonzero
integral
elements
[i.e.,
the
n
→
+∞]
of
the
elements
a
∈
k
×
such
that
a
−n
fails
to
converge
to
0,
as
N
gp
arithmetic
data,
the
associated
groupification
M
of
the
arithmetic
data
M
,
the
subgroup
of
invertible
elements
M
×
of
the
arithmetic
data
M
,
or
the
maximal
compact
subgroups
of
the
subgroups
of
the
arithmetic
data
obtained
as
subgroups
of
invariants
for
various
open
subgroups
of
the
Galois
group.
Finally,
we
shall
write
TG
for
the
category
of
topological
groups
and
continuous
homomorphisms
and
TG
⊇
TG
hyp
⊇
TG
sB
for
the
subcategories
determined,
respectively,
by
the
étale
fundamental
groups
of
arbitrary
hyperbolic
orbicurves
over
MLF’s
and
the
étale
fundamental
groups
of
hy-
perbolic
orbicurves
of
strictly
Belyi
type
over
MLF’s,
and
the
homomorphisms
that
induce
open
injections
on
the
quotients
constituted
by
the
absolute
Galois
groups
of
the
base
field
MLF’s;
also,
we
shall
use
the
same
notation,
except
with
“TG”
re-
placed
by
TG
to
denote
the
various
subcategories
determined
by
the
isomorphisms.
Thus,
for
T
∈
{TF,
TCG,
TLG,
TM,
TS,
TS},
the
assignment
(Π
M
)
→
Π
determines
various
compatible
natural
functors
C
T
MLF
→
TG
[as
well
as
double
underlined
versions
of
these
functors].
(iv)
Observe
that
[in
the
notation
of
(i)]
the
field
structure
of
k
determines,
via
the
inverse
morphism
to
log
k
,
a
structure
of
topological
field
on
the
topological
group
k
∼
.
Since
the
various
operations
applied
here
to
construct
this
field
struc-
ture
on
k
∼
[such
as,
for
instance,
the
power
series
used
to
define
log
k
]
are
clearly
intrinsically
defined
[cf.
the
natural
functors
defined
in
(iii)],
we
thus
obtain
that
the
construction
that
assigns
(the
ind-topological
field
k,
with
its
natural
Π
k
-action)
→
(the
ind-topological
field
k
∼
,
with
its
natural
Π
k
-action)
determines
a
natural
functor
MLF
MLF
log
TF,TF
:
C
TF
→
C
TF
—
which
we
shall
refer
to
as
the
log-Frobenius
functor
[cf.
Remark
3.6.2
below].
Since
log
k
determines
a
functorial
isomorphism
between
the
fields
k,
k
∼
,
it
follows
TOPICS
IN
ABSOLUTE
ANABELIAN
GEOMETRY
III
69
immediately
that
the
functor
log
TF,TF
is
isomorphic
to
the
identity
functor
[hence,
in
particular,
is
an
equivalence
of
categories].
By
composing
log
TF,TF
with
the
various
natural
functors
defined
in
(iii),
we
also
obtain,
for
T
∈
{TLG,
TCG,
TM},
a
functor
MLF
log
TF,T
:
C
TF
→
C
T
MLF
—
which
[by
abuse
of
terminology]
we
shall
also
refer
to
as
“the
log-Frobenius
functor”.
In
a
similar
vein,
the
assignments
(the
ind-topological
field
k,
with
its
natural
Π
k
-action)
×
→
(the
ind-topological
space
k
,
with
its
natural
Π
k
-action)
(the
ind-topological
field
k,
with
its
natural
Π
k
-action)
×
→
(the
ind-topological
space
(k
)
pf
,
with
its
natural
Π
k
-action)
determine
natural
functors
MLF
MLF
→
C
TS
;
λ
×
:
C
TF
MLF
MLF
λ
×pf
:
C
TF
→
C
TS
together
with
diagrams
of
functors
MLF
C
TF
⏐
⏐
×pf
λ
MLF
C
TS
log
TF,TF
−→
MLF
C
TF
⏐
⏐
×
λ
MLF
C
TF
⏐
ι
⏐
⏐
×
⏐
×
λ
λ
×pf
=
MLF
C
TS
MLF
C
TS
ι
log
—
where
we
write
ι
log
:
λ
×
◦log
TF,TF
→
λ
×pf
for
the
natural
transformation
induced
×
by
the
natural
inclusion
“(k
∼
)
×
→
k
∼
=
(O
×
)
pf
→
(k
)
pf
”
and
ι
×
:
λ
×
→
λ
×pf
k
×
×
for
the
natural
transformation
induced
by
the
natural
map
“k
→
(k
)
pf
”.
Finally,
we
note
that
the
subfield
of
Galois-invariants
“(k
∼
)
Π
k
”
of
the
field
“k
∼
”
obtained
by
the
above
construction
[i.e.,
the
arithmetic
data
of
an
object
in
the
image
of
the
log-Frobenius
functor
log
TF,TF
]
is
equipped
with
a
natural
“compactum”
—
i.e.,
the
compact
submodule
of
k
∼
=
(O
×
)
pf
determined
by
the
image
of
the
subgroup
k
O
k
×
=
(O
×
)
Π
k
⊆
O
×
of
Galois-invariants
of
O
×
—
which
we
shall
refer
to
as
the
k
k
k
pre-log-shell
λ
(ΠM
)
⊆
log
arith
TF,TF
((Π
M
))
MLF
)]
of
the
arithmetic
data
log
arith
[where
(Π
M
)
∈
Ob(C
TF
TF,TF
((Π
M
))
of
the
object
determined
by
applying
the
log-Frobenius
functor
log
TF,TF
to
the
object
κ
(Π
M
).
(v)
In
the
notation
of
(i),
suppose
further
that
T
∈
{TLG,
TCG,
TM};
let
(Π
M
)
be
an
MLF-Galois
T-pair.
Then
we
shall
refer
to
the
profinite
Π-module
def
(M
)
=
Hom(Q/Z,
M
)
μ
Z
as
the
cyclotome
associated
to
(Π
M
).
Also,
we
shall
[which
is
isomorphic
to
Z]
def
(M
)
⊗
Q/Z.
write
μ
Q/Z
(M
)
=
μ
Z
70
SHINICHI
MOCHIZUKI
(vi)
Recall
the
“image
via
the
Kummer
map
of
the
multiplicative
group
of
an
algebraic
closure
of
the
base
field”
×
1
k
→
lim
(Π))
−→
H
(J,
μ
Z
J
[where
“J”
ranges
over
the
open
subgroups
of
Π]
—
which
was
constructed
via
a
purely
“group-theoretic”
algorithm
in
Corollary
1.10,
(d),
(h),
for
Π
∈
Ob(TG
sB
).
Write
Anab
for
the
category
whose
objects
are
pairs
×
1
Π,
Π
{k
→
lim
H
(J,
μ
(Π))}
−→
Z
J
consisting
of
an
object
Π
∈
Ob(TG
sB
),
together
with
the
image
of
the
Kummer
map
reviewed
above,
equipped
with
its
topological
field
structure
and
natural
action
via
Π
—
all
of
which
is
to
be
understood
as
constructed
via
the
“group-theoretic”
algo-
rithms
of
Corollary
1.10,
(d),
(h)
[cf.
Remark
3.1.2
below]
—
and
whose
morphisms
are
the
morphisms
induced
by
isomorphisms
of
TG
sB
.
Thus,
we
obtain
a
natural
functor
κ
An
TG
sB
−→
Anab
which
[as
is
easily
verified]
is
an
equivalence
of
categories,
a
quasi-inverse
for
which
is
given
by
the
natural
projection
functor
Anab
→
TG
sB
.
Remark
3.1.1.
Observe
that
[in
the
notation
of
Definition
3.1,
(i)]
the
topology
on
the
field
k,
the
groups
k
×
and
O
k
×
,
or
the
monoid
O
k
is
completely
determined
by
the
field,
group,
or
monoid
structures
of
these
objects.
Indeed,
the
topology
on
O
k
×
is
precisely
the
profinite
topology;
the
topologies
on
k,
k
×
,
and
O
k
are
determined
by
the
topology
on
the
subset
O
k
×
⊆
O
k
⊆
k
×
⊆
k
[cf.
the
various
natural
functors
of
Definition
3.1,
(iii);
the
fact
that
O
k
×
⊆
k
×
may
be
characterized
as
the
subgroup
of
elements
divisible
by
arbitrary
powers
of
some
prime
number].
Suppose
that
T
=
TS,
TS.
Then
note
that
one
may
apply
this
observation
to
the
various
subfields,
subgroups,
or
submonoids
obtained
from
the
arithmetic
data
of
an
MLF-Galois
T-pair
by
taking
the
invariants
with
respect
to
some
open
subgroup
of
the
Galois
group.
Thus,
we
conclude
that
one
obtains
an
entirely
equivalent
theory
if
one
omits
the
specification
of
the
topology,
as
well
as
of
the
“ind-”
structure
[i.e.,
one
works
with
the
inductive
limit
fields,
groups,
or
monoids,
as
opposed
to
the
inductive
systems
of
such
objects]
from
the
objects
of
T
considered
in
Definition
MLF
is
precisely
the
data
3.1,
(i).
In
particular,
the
data
that
forms
an
object
of
C
TM
used
to
construct
the
“model
p-adic
Frobenioids”
of
[Mzk17],
Example
1.1.
Remark
3.1.2.
It
is
important
to
note
that,
by
definition,
the
algorithms
of
Corollary
1.10
form
an
essential
portion
of
each
object
of
the
category
Anab.
Put
another
way,
the
“software”
constituted
by
these
algorithms
is
not
just
executed
once,
leaving
behind
some
“output
data”
that
suffices
for
the
remainder
of
the
development
of
the
theory,
but
rather
executed
over
and
over
again
within
each
object
of
Anab.
TOPICS
IN
ABSOLUTE
ANABELIAN
GEOMETRY
III
71
Remark
3.1.3.
One
natural
variant
of
the
notion
of
an
“MLF-Galois
T-pair
of
hyperbolic
orbicurve
type”
is
the
notion
of
an
“MLF-Galois
T-pair
of
tempered
hyperbolic
orbicurve
type”,
i.e.,
the
case
where
[in
the
notation
of
Definition
3.1,
(ii)]
Π
k
G
k
arises
from
the
tempered
fundamental
group
of
a
hyperbolic
orbicurve
over
k
[cf.
Remarks
1.9.1,
1.10.2].
We
leave
to
the
reader
the
routine
details
of
developing
the
resulting
tempered
version
of
the
theory
to
follow.
Proposition
3.2.
(Monoid
Cyclotomes
and
Kummer
Maps)
Let
T
∈
{TM,
TF};
(Π
M
T
)
∈
Ob(C
MLF
)
an
MLF-Galois
T-pair,
with
arithmetic
Ga-
T
lois
group
Π
G.
Write
(Π
M
TM
)
∈
Ob(C
MLF
TM
)
for
the
object
obtained
from
(Π
M
T
)
by
applying
the
identity
functor
if
T
=
TM
or
by
applying
the
natural
functor
of
Definition
3.1,
(iii),
if
T
=
TF.
Then:
(i)
The
arguments
given
in
the
proof
of
[Mzk9],
Proposition
1.2.1,
(vii),
yield
a
functorial
[i.e.,
relative
to
C
MLF
,
in
the
evident
sense
—
cf.
Remark
3.2.2
below]
T
algorithm
for
constructing
the
natural
isomorphism
∼
(M
TM
))
→
Z
H
2
(G,
μ
Z
—
i.e.,
by
composing
the
natural
isomorphism
[of
“Brauer
groups”]
∼
gp
)
H
2
(G,
μ
Q/Z
(M
TM
))
→
H
2
(G,
M
TM
[where
“gp”
denotes
the
groupification
of
a
monoid]
with
the
inverse
of
the
natural
isomorphism
[of
“Brauer
groups”]
∼
gp
unr
gp
)
)
→
H
2
(G,
M
TM
)
H
2
(G
unr
,
(M
TM
unr
[where
M
TM
⊆
M
TM
denotes
the
submonoid
of
elements
fixed
by
the
kernel
of
the
quotient
G
G
unr
of
Corollary
1.10,
(b)]
followed
by
the
natural
composite
isomorphism
∼
unr
gp
∼
unr
gp
unr
×
∼
Z)
→
H
2
(G
unr
,
(M
TM
)
)
→
H
2
(G
unr
,
(M
TM
)
/(M
TM
)
)
→
H
2
(
Z,
Q/Z
[where
“×”
denotes
the
subgroup
of
invertible
elements
of
a
monoid;
the
isomor-
unr
gp
unr
×
∼
)
/(M
TM
)
→
Z
is
obtained
by
considering
a
generator
of
the
phism
(M
TM
∼
unr
unr
×
∼
monoid
M
TM
/(M
TM
)
=
N;
we
apply
the
isomorphism
G
unr
→
Z
of
Corollary
1.10,
(b)]
and
then
applying
the
functor
Hom(Q/Z,
−)
to
the
resulting
isomorphism
∼
H
2
(G,
μ
Q/Z
(M
TM
))
→
Q/Z
[cf.
also
Remark
3.2.1
below].
(ii)
By
considering
the
action
of
open
subgroups
H
⊆
Π
on
elements
of
M
TM
H
that
are
roots
of
elements
of
M
TM
[i.e.,
the
submonoid
of
M
TM
consisting
of
H-
invariant
elements],
we
obtain
a
functorial
[i.e.,
relative
to
C
MLF
,
in
the
evident
T
sense]
algorithm
for
constructing
the
Kummer
maps
H
M
TM
→
H
1
(H,
μ
(M
TM
));
Z
1
M
TM
→
lim
(M
TM
))
−→
H
(J,
μ
Z
J
—
where
“J”
ranges
over
the
open
subgroups
of
Π.
In
particular,
the
“μ
(M
TM
)”
in
Z
(G)”
[cf.
Remarks
3.2.1,
3.2.2
below];
if,
the
above
display
may
be
replaced
by
“μ
Z
72
SHINICHI
MOCHIZUKI
moreover,
(Π
M
T
)
is
of
hyperbolic
orbicurve
type,
then
the
“μ
(M
TM
)”
in
Z
(Π)”
[cf.
Corollary
1.10,
(c)]
or
“μ
κ
(Π)”
the
above
display
may
be
replaced
by
“μ
Z
Z
[cf.
Remark
1.10.3,
(ii)]
—
cf.
Remark
3.2.2
below.
(iii)
Suppose
that
(Π
M
T
)
is
of
strictly
Belyi
type.
Then
the
construction
of
Corollary
1.10,
(h),
determines
an
additive
structure
[hence,
in
particular,
a
topological
field
structure]
on
the
union
with
“{0}”
of
the
group
generated
by
the
image
of
the
Kummer
map
1
(M
TM
))
M
TM
→
lim
−→
H
(J,
μ
Z
J
,
of
(ii).
In
particular,
these
constructions
yield
a
functorial
[i.e.,
relative
to
C
MLF
T
in
the
evident
sense
—
cf.
Remark
3.2.2
below]
algorithm
for
constructing
this
topological
field
structure.
),
then
the
natural
functor
of
Definition
3.1,
(iv)
If
(Π
∗
M
T
∗
)
∈
Ob(C
MLF
T
(iii),
induces
an
injection
Isom
C
MLF
((Π
M
T
),
(Π
∗
M
T
∗
))
→
Isom
TG
(Π,
Π
∗
)
T
on
sets
of
isomorphisms;
this
injection
is
a
bijection
if
T
=
TM,
or
if
[T
is
either
TM
or
TF,
and]
(Π
M
T
),
(Π
∗
M
T
∗
)
are
of
strictly
Belyi
type.
In
particular,
if
(Π
M
T
)
is
of
hyperbolic
orbicurve
type,
then
the
group
Aut
C
MLF
((Π
M
T
))
T
—
which
is
isomorphic
to
a
subgroup
of
Aut
TG
(Π)
that
contains
the
subgroup
of
Aut
TG
(Π)
determined
by
the
inner
automorphisms
of
Π
—
is
center-free;
the
categories
TG
hyp
,
TG
sB
,
TG
hyp
,
TG
sB
,
C
MLF-hyp
,
C
MLF-sB
,
C
MLF-hyp
,
C
MLF-sB
are
T
T
T
T
id-rigid
[cf.
§0].
(v)
The
algorithm
of
(iii)
yields
a
natural
[1-]factorization
MLF-sB
C
TF
−→
C
T
MLF-sB
log
T,T
−→
C
T
MLF-sB
—
where
T
∈
{TF,
TLG,
TCG,
TM};
the
first
arrow
is
the
natural
functor
of
Def-
inition
3.1,
(iii),
if
T
=
TM,
or
the
identity
functor
if
T
=
TF
—
of
the
[“sB”
MLF
→
C
T
MLF
of
Definition
versions
of
the]
log-Frobenius
functors
log
TF,T
:
C
TF
3.1,
(iv).
Moreover,
the
functor
log
T,T
is
isomorphic
to
the
identity
functor
[hence,
in
particular,
is
an
equivalence
of
categories].
Proof.
Assertions
(i),
(ii),
(iii),
(v)
are
immediate
from
the
constructions
that
appear
in
the
statement
of
these
assertions
[together
with
the
references
quoted
in
these
constructions].
The
injectivity
portion
of
assertion
(iv)
follows
from
the
functorial
algorithms
of
assertions
(i),
(ii)
[which
imply
that
automorphisms
of
(Π
M
TM
)
that
act
trivially
on
Π
necessarily
act
trivially
on
M
TM
].
In
light
of
this
injectivity,
the
center-free-ness
portion
of
assertion
(iv)
follows
immediately
from
the
slimness
of
Π
[cf.,
e.g.,
[Mzk20],
Proposition
2.3,
(ii)].
The
surjectivity
portion
of
assertion
(iv)
follows
from
assertion
(iii),
when
(Π
M
T
),
(Π
∗
M
T
∗
)
TOPICS
IN
ABSOLUTE
ANABELIAN
GEOMETRY
III
73
∼
are
of
strictly
Belyi
type,
and
from
considering
the
“copy
of
M
TM
→
O
embedded
k
∼
in
abelianizations
of
open
subgroups
of
G
→
G
k
via
local
class
field
theory”
[cf.,
e.g.,
[Mzk9],
Proposition
1.2.1,
(iii),
(iv)],
together
with
assertions
(i),
(ii)
[cf.
also
the
first
displayed
isomorphism
of
Corollary
1.10,
(b)],
when
T
=
TM.
Remark
3.2.1.
Note
that
the
algorithm
applied
to
construct
the
natural
isomor-
phism
of
Corollary
1.10,
(a),
is
essentially
the
same
as
the
algorithm
of
Proposition
3.2,
(i).
In
particular,
this
algorithm
does
not
require
that
(Π
M
T
)
be
of
hy-
perbolic
orbicurve
type.
Thus,
by
imposing
the
condition
of
“compatibility
with
the
natural
isomorphism
of
Corollary
1.10,
(a)”,
we
thus
obtain,
in
the
context
of
Proposition
3.2,
(i),
a
functorial
algorithm
for
constructing
the
natural
isomorphism
∼
μ
(M
TM
)
→
μ
(G)
Z
Z
[cf.
also
Remark
1.10.3,
(ii)].
Remark
3.2.2.
Note
that
[cf.
Remark
1.10.1,
(iii)]
the
functoriality
of
Propo-
∼
sition
3.2,
(i),
when
applied
to
the
isomorphism
H
2
(G,
μ
(M
TM
))
→
Z,
is
to
be
Z
that
ap-
understood
in
the
sense
of
a
“compatibility”,
relative
to
dividing
the
“
Z”
pears
as
the
codomain
of
these
isomorphisms
by
a
factor
given
by
the
index
of
the
image
of
the
induced
open
homomorphism
on
arithmetic
Galois
groups
[cf.
Defini-
tion
3.1,
(ii)].
A
similar
remark
[cf.
Remark
1.10.1,
(i)]
applies
to
the
cyclotome
(Π)”
that
appears
in
Proposition
3.2,
(ii).
We
leave
the
routine
details
to
the
“μ
Z
reader.
In
a
similar
vein,
one
may
consider
Kummer
maps
for
“O
×
”
[as
opposed
to
“O
”],
in
which
case
the
natural
isomorphism
of
Remark
3.2.1
is
only
determined
×
-multiple
[cf.
[Mzk17],
Remark
2.4.2].
up
to
a
Z
Proposition
3.3.
(Unit
Kummer
Maps)
Let
T
∈
{TLG,
TCG}.
Let
(Π
MLF
M
)
∈
Ob(C
T
)
be
an
MLF-Galois
T-pair,
with
arithmetic
Galois
group
Π
G.
Then:
(i)
By
considering
the
action
of
open
subgroups
H
⊆
Π
on
elements
of
M
that
are
roots
of
elements
of
M
H
[i.e.,
the
subgroup
of
M
consisting
of
H-invariant
elements],
we
obtain
a
functorial
[i.e.,
relative
to
C
MLF
,
in
the
evident
sense]
T
algorithm
for
constructing
the
Kummer
maps
(M
));
M
H
→
H
1
(H,
μ
Z
1
M
→
lim
(M
))
−→
H
(J,
μ
Z
J
—
where
“J”
ranges
over
the
open
subgroups
of
Π.
In
this
situation
[unlike
the
∼
situation
of
Proposition
3.2,
(ii)],
the
natural
isomorphism
μ
(M
)
→
μ
(G)
[cf.
Z
Z
×
-)multiple
if
Remark
3.2.1]
is
only
determined
up
to
a
{±1}-
(respectively,
Z
T
=
TLG
(respectively,
T
=
TCG)
[cf.
(ii)
below;
[Mzk17],
Remark
2.4.2,
in
the
case
T
=
TCG].
74
SHINICHI
MOCHIZUKI
∼
(ii)
If
(Π
∗
M
∗
)
∈
Ob(C
MLF
),
then
any
isomorphism
(Π
M
)
→
(Π
∗
T
∼
∼
M
∗
)
induces
isomorphisms
Π
→
Π
∗
,
μ
(M
)
→
μ
(M
∗
),
which
determine
an
in-
Z
Z
jection
((Π
M
),
(Π
∗
M
∗
))
→
Isom
TG
(Π,
Π
∗
)
×
Isom
TG
(μ
(M
),
μ
(M
∗
))
Isom
C
MLF
Z
Z
T
—
which
is
a
bijection
if
T
=
TCG.
If
T
=
TLG,
then
the
homomorphism
((Π
M
),
(Π
∗
M
∗
))
→
Isom
TG
(Π,
Π
∗
)
is
surjective,
with
fibers
of
Isom
C
MLF
T
cardinality
two.
Proof.
The
portion
of
assertion
(i)
concerning
Kummer
maps
is
immediate
from
the
definitions
and
the
references
quoted.
The
portion
of
assertion
(i)
concerning
∼
(M
)
→
μ
(G)
follows
by
observing
that
the
algorithm
of
Propo-
the
isomorphism
μ
Z
Z
sition
3.2,
(i)
[cf.
also
Remark
3.2.1]
may
be
applied,
up
to
a
{±1}-
(respectively,
×
-)indeterminacy,
if
T
=
TLG
(respectively,
T
=
TCG).
The
injectivity
portion
Z
of
assertion
(ii)
follows
from
assertion
(i)
via
a
similar
argument
to
the
argument
used
to
derive
the
injectivity
portion
of
Proposition
3.2,
(iv),
from
Proposition
3.2,
(i),
(ii);
the
surjectivity
onto
Isom
TG
(Π,
Π
∗
)
follows
from
a
similar
argument
to
the
argument
applied
to
prove
the
surjectivity
portion
of
Proposition
3.2,
(iv).
If
T
=
TCG
(respectively,
T
=
TLG),
then
the
remainder
of
assertion
(ii)
follows
by
×
on
M
(respectively,
observing
that
observing
that
there
is
a
natural
action
of
Z
×
as
soon
as
an
automorphism
of
(Π
M
)
preserves
the
submonoid
O
⊆
k
∼
=
M
k
×
[i.e.,
preserves
the
“positive
elements”
of
k
/O
×
∼
=
Q],
one
may
apply
the
functorial
algorithm
of
Proposition
3.2,
(i)).
k
Lemma
3.4.
(Topological
Distinguishability
of
Additive
and
Mul-
∼
tiplicative
Structures)
In
the
notation
of
Definition
3.1,
(i),
let
α
:
k
×
→
k
×
be
an
automorphism
of
the
topological
group
k
×
,
α
pf
:
(k
×
)
pf
→
(k
×
)
pf
the
automorphism
induced
on
the
perfection.
Then
α
pf
((O
k
)
pf
)
⊆
(O
k
×
)
pf
.
Proof.
Indeed,
since
O
k
×
is
easily
verified
to
be
the
maximal
compact
subgroup
of
∼
k
×
,
α
induces
an
isomorphism
O
k
×
→
O
k
×
.
Thus,
α
pf
((O
k
×
)
pf
)
=
(O
k
×
)
pf
,
so
an
in-
clusion
α
pf
((O
k
)
pf
)
⊆
(O
k
×
)
pf
would
imply
that
(O
k
)
pf
⊆
(O
k
×
)
pf
,
a
contradiction.
Definition
3.5.
(i)
We
shall
refer
to
as
a
diagram
of
categories
D
=
(
Γ
D
,
{D
v
},
{D
e
})
any
collection
of
data
as
follows:
(a)
an
oriented
graph
Γ
D
[cf.
§0];
(b)
for
each
vertex
v
of
Γ
D
,
a
category
D
v
;
(c)
for
each
edge
e
of
Γ
D
that
runs
from
a
vertex
v
1
to
a
vertex
v
2
,
a
functor
D
e
:
D
v
1
→
D
v
2
.
Let
D
=
(
Γ
D
,
{D
v
},
{D
e
})
be
a
diagram
of
categories.
Then
observe
that
any
path
[γ]
[cf.
§0]
on
Γ
D
that
runs
from
a
vertex
v
1
to
a
vertex
v
2
determines
—
i.e.,
TOPICS
IN
ABSOLUTE
ANABELIAN
GEOMETRY
III
75
by
composing
the
various
functors
“D
e
”,
for
edges
e
that
appear
in
this
path
—
a
functor
D
[γ]
:
D
v
1
→
D
v
2
.
We
shall
refer
to
the
diagram
of
categories
E
obtained
by
restricting
the
data
of
D
to
an
oriented
subgraph
Γ
E
of
Γ
D
as
a
subdiagram
of
categories
of
D.
(ii)
Let
D
=
(
Γ
D
,
{D
v
},
{D
e
})
be
a
diagram
of
categories.
Then
we
shall
refer
to
as
a
family
of
homotopies
H
=
(E
H
,
{ζ
})
on
D
any
collection
of
data
as
follows:
(a)
a
saturated
[cf.
§0]
set
E
H
⊆
Ω(
Γ
D
)
×
Ω(
Γ
D
)
of
ordered
pairs
of
paths
on
Γ
D
,
which
we
shall
refer
to
as
the
boundary
set
of
the
family
of
homotopies
H;
we
shall
refer
to
every
path
on
Γ
D
that
occurs
as
a
component
of
an
element
of
E
H
as
a
boundary
set
path;
(b)
for
each
=
([γ
1
],
[γ
2
])
∈
E
H
,
a
natural
transformation
ζ
:
D
[γ
1
]
→
D
[γ
2
]
—
which
we
shall
refer
to
as
a
homotopy
from
[γ
1
]
to
[γ
2
]
—
such
that
the
following
conditions
are
satisfied:
ζ
([γ],[γ])
is
the
identity
natural
transformation
for
each
([γ],
[γ])
∈
E
H
;
if
=
([γ
1
],
[γ
2
]),
=
([γ
2
],
[γ
3
]),
and
=
([γ
1
],
[γ
3
])
belong
to
E
H
,
then
ζ
=
ζ
◦
ζ
;
if,
for
some
[γ
3
],
[γ
4
]
∈
Ω(
Γ
D
),
the
pairs
=
([γ
1
],
[γ
2
])
and
=
([γ
3
]◦[γ
1
]◦[γ
4
],
[γ
3
]◦
[γ
2
]
◦
[γ
4
])
belong
to
E
H
,
then
ζ
=
D
[γ
3
]
◦
ζ
◦
D
[γ
4
]
.
If,
in
this
situation,
E
H
is
the
smallest
[cf.
§0]
saturated
subset
of
Ω(
Γ
D
)
×
Ω(
Γ
D
)
∗
⊆
E
H
,
then
we
shall
say
that
the
family
of
ho-
that
contains
a
given
subset
E
H
∗
.
We
motopies
H
=
(E
H
,
{ζ
})
is
generated
by
the
homotopies
indexed
by
E
H
shall
refer
to
a
family
of
homotopies
H
=
(E
H
,
{ζ
})
on
D
as
symmetric
if
E
H
is
symmetrically
saturated
[cf.
§0].
[Thus,
if
H
=
(E
H
,
{ζ
})
is
symmetric,
then
every
ζ
is
an
isomorphism.]
We
shall
refer
to
a
collection
of
families
of
homo-
topies
{H
ι
=
(E
H
ι
,
{ζ
ι
ι
})}
ι∈I
on
D
as
being
compatible
if
there
exists
a
family
of
homotopies
H
=
(E
H
,
{ζ
})
on
D
such
that,
for
each
ι
∈
I,
ι
∈
E
H
ι
,
we
have
E
H
ι
⊆
E
H
and
ζ
ι
ι
=
ζ
ι
.
(iii)
Let
D
=
(
Γ
D
,
{D
v
},
{D
e
})
be
a
diagram
of
categories.
Then
we
shall
refer
to
as
an
observable
S
=
(S,
v
S
,
H)
[on
D]
any
collection
of
data
as
follows:
(a)
a
diagram
of
categories
S
=
(
Γ
S
,
{S
v
},
{S
e
})
that
contains
D
as
a
subdi-
agram
of
categories
[so
Γ
D
⊆
Γ
S
];
(b)
a
vertex
v
S
of
Γ
S
,
which
we
shall
refer
to
as
the
observation
vertex,
such
that
the
set
of
vertices
of
Γ
S
\
Γ
D
is
equal
to
{v
S
},
and,
moreover,
every
edge
of
Γ
S
\
Γ
D
runs
from
a
vertex
of
Γ
D
to
v
S
;
(c)
a
family
of
homotopies
H
on
S
such
that
every
boundary
set
path
of
H
has
terminal
vertex
equal
to
v
S
.
Let
S
=
(S,
v
S
,
H)
be
an
observable
on
D.
Then
we
shall
say
that
S
is
symmetric
if
H
is
symmetric.
We
shall
say
that
S
is
a
core
[on
D]
if
the
boundary
set
of
H
is
equal
to
the
set
of
all
co-verticial
pairs
of
paths
on
the
underlying
oriented
graph
Γ
S
of
S
with
terminal
vertex
equal
to
v
S
[which
implies
that
S
is
symmetric],
and,
moreover,
every
vertex
of
Γ
D
appears
as
the
initial
vertex
of
a
path
on
Γ
S
with
terminal
vertex
equal
to
v
S
.
Suppose
that
S
is
a
core
on
D.
Then
we
shall
refer
to
76
SHINICHI
MOCHIZUKI
the
observation
vertex
of
S
as
the
core
vertex
of
S
and
[by
abuse
of
terminology,
when
there
is
no
fear
of
confusion]
to
S
v
S
as
a
“core
on
D”.
(iv)
Let
S
=
(S,
v
S
,
H)
be
a
core
on
a
diagram
of
categories
D
=
(
Γ
D
,
{D
v
},
{D
e
}).
Then
we
shall
refer
to
as
a
telecore
T
=
(T
,
J
)
on
D
over
the
core
S
any
collection
of
data
as
follows:
(a)
a
diagram
of
categories
T
=
(
Γ
T
,
{T
v
},
{T
e
})
that
contains
S
as
a
subdi-
agram
of
categories
[so
Γ
D
⊆
Γ
S
⊆
Γ
T
]
such
that
Γ
T
,
Γ
S
have
the
same
vertices,
and,
moreover,
every
edge
of
Γ
T
\
Γ
S
runs
from
v
S
to
a
vertex
of
Γ
D
;
we
shall
refer
to
such
edges
of
Γ
T
as
the
telecore
edges;
(b)
J
is
a
family
of
homotopies
on
T
such
that
J
|
S
=
H
whose
boundary
set
is
equal
to
the
subset
of
Ω(
Γ
T
)
of
pairs
([γ
3
]
◦
[γ
1
],
[γ
3
]
◦
[γ
2
]),
where
([γ
1
],
[γ
2
])
is
a
co-verticial
pair
of
paths
on
Γ
T
with
terminal
vertex
equal
to
v
S
,
and
[γ
3
]
is
a
path
on
Γ
T
with
initial
vertex
equal
to
v
S
.
In
this
situation,
a
family
of
homotopies
H
cnct
on
T
that
is
compatible
with
J
will
be
referred
to
as
a
contact
structure
for
the
telecore
T
.
(v)
Let
D
=
(
Γ
D
,
{D
v
},
{D
e
})
and
D
=
(
Γ
D
,
{D
v
},
{D
e
})
be
diagrams
of
categories.
Then
a
1-morphism
of
diagrams
of
categories
Φ
:
D
→
D
is
defined
to
be
a
collection
of
data
follows:
(a)
a
morphism
of
oriented
graphs
Φ
Γ
:
Γ
D
→
Γ
D
;
(b)
for
each
vertex
v
of
Γ
D
,
a
functor
Φ
v
:
D
v
→
D
Φ
Γ
(v)
;
(c)
for
each
edge
e
of
Γ
D
that
runs
from
a
vertex
v
1
to
a
vertex
v
2
,
an
∼
isomorphism
of
functors
Φ
e
:
D
Φ
Γ
(e)
◦
Φ
v
1
→
Φ
v
2
◦
D
e
.
A
2-morphism
Θ
:
Φ
→
Ψ
between
1-morphisms
Φ,
Ψ
:
D
→
D
such
that
Φ
Γ
=
Ψ
Γ
is
defined
to
be
a
collection
of
natural
transformations
{Θ
v
:
Φ
v
→
Ψ
v
},
where
v
ranges
over
the
vertices
of
Γ
D
,
such
that
Ψ
e
◦
(D
Φ
Γ
(e)
◦
Θ
v
1
)
=
(Θ
v
2
◦
D
e
)
◦
Φ
e
:
D
Φ
Γ
(e)
◦
Φ
v
1
→
Ψ
v
2
◦
D
e
for
each
edge
e
of
Γ
D
that
runs
from
a
vertex
v
1
to
a
vertex
v
2
.
We
shall
say
that
a
1-morphism
Φ
:
D
→
D
is
an
equivalence
of
diagrams
of
categories
if
there
exists
a
1-morphism
Ψ
:
D
→
D
such
that
Ψ
◦
Φ,
Φ
◦
Ψ
are
[2-]isomorphic
to
the
respective
identity
1-morphisms
of
D,
D
.
If
D
(respectively,
D
)
is
equipped
with
a
family
of
homotopies
H
(respectively,
H
),
then
we
shall
say
that
an
equivalence
∼
Φ
:
D
→
D
is
compatible
with
H,
H
if
Φ
Γ
induces
a
bijection
between
the
boundary
sets
of
H,
H
,
and,
moreover,
the
natural
transformations
that
constitute
H,
H
[cf.
the
data
of
(ii),
(b)]
are
compatible
[in
the
evident
sense]
with
the
natural
transformations
that
constitute
Φ
[cf.
the
data
(c)
in
the
above
definition];
in
this
situation,
one
verifies
immediately
that
if
Φ
is
compatible
with
H,
H
,
then
so
is
∼
any
equivalence
Ψ
:
D
→
D
that
is
isomorphic
to
Φ.
We
shall
say
that
D
is
vertex-
rigid
(respectively,
edge-rigid)
if,
for
every
vertex
v
(respectively,
edge
e)
of
Γ
D
,
the
TOPICS
IN
ABSOLUTE
ANABELIAN
GEOMETRY
III
77
category
D
v
(respectively,
the
functor
D
e
)
is
id-rigid
(respectively,
rigid)
[cf.
§0].
If
D
is
vertex-rigid
and
edge-rigid,
then
we
shall
say
that
D
is
totally
rigid.
Thus,
∼
if
D
is
edge-rigid,
then
any
equivalence
Φ
:
D
→
D
is
completely
determined
by
Φ
Γ
and
the
{Φ
v
}
[i.e.,
the
data
(a),
(b)
in
the
above
definition].
In
a
similar
vein,
∼
if
D
is
vertex-rigid,
then
any
two
isomorphic
equivalences
Φ,
Ψ
:
D
→
D
admit
a
∼
unique
[2-]isomorphism
Φ
→
Ψ.
In
particular,
if
D
is
vertex-rigid,
then
it
is
natural
to
speak
of
the
automorphism
group
Aut(D)
of
D,
i.e.,
the
group
determined
by
the
isomorphism
classes
of
self-equivalences
of
D.
(vi)
Let
D
=
(
Γ
D
,
{D
v
},
{D
e
})
be
a
diagram
of
categories;
a
nexus
of
Γ
D
[cf.
§0];
D
≤
,
D
≥
the
subdiagrams
of
categories
determined,
respectively,
by
the
pre-
and
post-nexus
portions
of
Γ
D
[cf.
§0].
Then
we
shall
say
that
D
is
totally
-rigid
if
the
pre-nexus
portion
D
≤
is
totally
rigid.
Let
us
suppose
that
D
is
totally
-rigid.
Write
Aut
(D
≤
)
⊆
Aut(D
≤
)
for
the
subgroup
of
isomorphism
classes
of
self-equivalences
of
D
≤
that
preserve
and
induce
a
self-equivalence
of
D
that
is
isomorphic
to
the
identity
self-
∼
equivalence.
Let
Φ
≤
:
D
≤
→
D
≤
be
a
self-equivalence
whose
isomorphism
class
∼
[Φ
≤
]
∈
Aut
(D
≤
).
Then
Φ
≤
extends
naturally
to
an
equivalence
Φ
:
D
→
D
which
is
the
identity
on
Γ
D
≥
and
which
associates
to
each
vertex
v
=
of
Γ
D
≥
the
identity
self-equivalence
of
D
v
.
[Here,
we
observe
that
the
isomorphism
of
func-
tors
of
(v),
(c),
is
naturally
determined
by
the
isomorphism
of
(Φ
≤
)
with
the
identity
self-equivalence
of
D
.]
Moreover,
this
assignment
Φ
≤
→
Φ
clearly
maps
isomorphic
equivalences
to
isomorphic
equivalences
and
is
compatible
with
composition
of
equivalences.
In
particular,
this
assignment
yields
a
natu-
ral
“action”
of
the
group
Aut
(D
≤
)
on
D.
We
shall
refer
to
the
resulting
self-
equivalences
of
D
as
nexus
self-equivalences
of
D
[relative
to
the
nexus
]
and
the
resulting
classes
of
self-equivalences
of
D
[i.e.,
arising
from
isomorphism
classes
of
“Φ
≤
”]
as
nexus-classes
of
self-equivalences
of
D
[relative
to
the
nexus
].
Remark
3.5.1.
If
one
just
works
with
diagrams
of
categories
without
considering
any
observables,
then
it
is
difficult
to
understand
the
“global
structure”
of
the
diagram
since
[by
definition!]
it
does
not
make
sense
to
speak
of
the
relationship
between
objects
that
belong
to
different
categories
[e.g.,
at
distinct
vertices
of
the
diagram].
Thus:
The
notion
of
an
observable
may
be
thought
of
as
a
sort
of
“partial
pro-
jection
of
the
dynamics
of
a
diagram
of
categories”
onto
a
single
category,
within
which
it
makes
sense
to
compare
objects
that
arise
from
distinct
categories
at
distinct
vertices
of
the
diagram.
Moreover:
A
core
on
a
diagram
of
categories
may
be
thought
of
as
an
extraction
of
a
certain
portion
of
the
data
of
the
objects
at
the
various
categories
in
the
78
SHINICHI
MOCHIZUKI
diagram
that
is
invariant
with
respect
to
the
“dynamics”
arising
from
the
application
of
the
various
functors
in
the
diagram.
Put
another
way,
one
may
think
of
a
core
as
a
sort
of
“constant
portion”
of
the
diagram
that
lies,
in
a
consistent
fashion,
“under
the
entire
diagram”
[cf.
the
use
of
the
term
“core”
in
the
theory
of
[Mzk11],
§2].
Then:
A
telecore
may
be
thought
of
as
a
sort
of
partial
section
—
i.e.,
given
by
the
telecore
edges
—
of
the
“structure
morphisms
to
the
core”
which
does
not
disturb
the
coricity
[i.e.,
the
property
of
being
a
core]
of
the
original
core.
Moreover,
although,
in
the
definition
of
a
telecore,
we
do
not
assume
the
existence
of
families
of
homotopies
that
guarantee
the
compatibility
of
applying
composites
of
functors
by
traveling
along
arbitrary
co-verticial
pairs
of
paths
emanating
from
the
core
vertex,
any
failure
of
such
a
compatibility
may
always
be
eliminated
—
in
a
fashion
reminiscent
of
a
“telescoping
sum”
—
by
projecting
back
down
to
the
core
vertex
[cf.
the
discussion
of
Remark
3.6.1,
(ii),
(c),
below].
Put
another
way,
one
may
think
of
a
telecore
as
a
device
that
satisfies
a
sort
of
“time
lag
compatibility”,
i.e.,
as
a
device
whose
“compatibility
apparatus”
does
not
go
into
operation
immediately,
but
only
after
a
certain
“time
lag”
[arising
from
the
necessity
to
travel
back
down
to
the
core
vertex].
Also,
for
more
on
the
meaning
of
cores
and
telecores,
we
refer
to
Remark
3.6.5
below.
The
terminology
of
Definition
3.5
makes
it
possible
to
formulate
the
first
main
result
of
the
present
§3.
Corollary
3.6.
(MLF-Galois-theoretic
Mono-anabelian
Log-Frobenius
Compatibility)
Write
def
;
X
=
C
MLF-sB
T
def
E
=
TG
sB
;
def
MLF-sB
N
=
C
TS
—
where
[in
the
notation
of
Definition
3.1]
T
∈
{TM,
TF}.
Consider
the
diagram
of
categories
D
log
log
−→
X
−→
X
...
...
X
⏐
⏐
id
id
−1
...
...
id
+1
X
⏐
⏐
⏐
⏐
λ
×
λ
×pf
N
⏐
⏐
E
⏐
⏐
κ
An
Anab
⏐
⏐
E
TOPICS
IN
ABSOLUTE
ANABELIAN
GEOMETRY
III
79
—
where
we
use
the
notation
“log”,
“λ
×
”,
“λ
×pf
”
for
the
evident
[double-underlined/
overlined]
restrictions
of
the
arrows
“log
T,T
”,
“λ
×
”,
“λ
×pf
”
of
Definition
3.1,
(iv)
[cf.
also
Proposition
3.2,
(iii),
(v)];
for
positive
integers
n
≤
6,
we
shall
denote
by
D
≤n
the
subdiagram
of
categories
of
D
determined
by
the
first
n
[of
the
six]
rows
of
D;
we
write
L
for
the
countably
ordered
set
determined
[cf.
§0]
by
the
infinite
linear
oriented
graph
Γ
opp
D
≤1
[so
the
elements
of
L
correspond
to
vertices
of
the
first
row
of
D]
and
L
†
=
L
∪
{}
def
for
the
ordered
set
obtained
by
appending
to
L
a
formal
symbol
[which
we
think
of
as
corresponding
to
the
unique
vertex
of
the
second
row
of
D]
such
that
<
,
for
all
∈
L;
id
denotes
the
identity
functor
at
the
vertex
∈
L;
the
notation
“.
.
.
”
denotes
an
infinite
repetition
of
the
evident
pattern.
Then:
(i)
For
n
=
4,
5,
6,
D
≤n
admits
a
natural
structure
of
core
on
D
≤n−1
.
That
is
to
say,
loosely
speaking,
E,
Anab
“form
cores”
of
the
functors
in
D.
(ii)
The
assignments
Π,
Π
{k
→
lim
(Π))}
→
(Π
O
k
),
−→
H
(J,
μ
Z
×
1
(Π
k
×
{0})
J
determine
[i.e.,
for
each
choice
of
T]
a
natural
“forgetful”
functor
Anab
φ
An
−→
X
which
is
an
equivalence
of
categories,
a
quasi-inverse
for
which
is
given
by
the
composite
π
An
:
X
→
Anab
of
the
natural
projection
functor
X
→
E
with
∼
κ
An
:
E
→
Anab;
write
η
An
:
φ
An
◦
π
An
→
id
X
for
the
isomorphism
arising
from
the
“group-theoretic”
algorithms
of
Corollary
1.10
[cf.
also
Proposition
3.2,
(ii),
(iii)].
Moreover,
φ
An
gives
rise
to
a
telecore
structure
T
An
on
D
≤4
,
whose
underlying
diagram
of
categories
we
denote
by
D
An
,
by
appending
to
D
≤5
telecore
edges
...
φ
+1
...
X
Anab
⏐
⏐
φ
log
−→
Anab
X
φ
−→
φ
−1
...
X
...
log
−→
X
from
the
core
Anab
to
the
various
copies
of
X
in
D
≤2
given
by
copies
of
φ
An
,
which
we
denote
by
φ
,
for
∈
L
†
.
That
is
to
say,
loosely
speaking,
φ
An
determines
a
0
]
for
the
path
telecore
structure
on
D
≤4
.
Finally,
for
each
∈
L
†
,
let
us
write
[β
1
on
Γ
D
An
of
length
0
at
and
[β
]
for
the
path
on
Γ
D
An
of
length
∈
{4,
5}
[i.e.,
depending
on
whether
or
not
=
]
that
starts
from
,
descends
[say,
via
λ
×
]
to
the
core
vertex
“Anab”,
and
returns
to
via
the
telecore
edge
φ
.
Then
the
collection
of
natural
transformations
−1
−1
{η
,
η
,
η
,
η
}
∈L,∈L
†
80
SHINICHI
MOCHIZUKI
—
where
we
write
η
for
the
identity
natural
transformation
from
the
arrow
φ
:
Anab
→
X
to
the
composite
arrow
id
◦
φ
:
Anab
→
X
and
∼
η
:
(D
An
)
[β
1
]
→
(D
An
)
[β
0
]
for
the
isomorphism
arising
from
η
An
—
generate
a
contact
structure
H
An
on
the
telecore
T
An
.
(iii)
The
natural
transformations
ι
log,
:
λ
×
◦
id
◦
log
→
λ
×pf
◦
id
+1
,
ι
×
:
λ
×
→
λ
×pf
[cf.
Definition
3.1,
(iv)]
belong
to
a
family
of
homotopies
on
D
≤3
that
determines
on
D
≤3
a
structure
of
observable
S
log
on
D
≤2
and,
moreover,
is
compatible
with
the
families
of
homotopies
that
constitute
the
core
and
telecore
structures
of
(i),
(ii).
(iv)
The
diagram
of
categories
D
≤2
does
not
admit
a
structure
of
core
on
D
≤1
which
[i.e.,
whose
constituent
family
of
homotopies]
is
compatible
with
[the
constituent
family
of
homotopies
of
]
the
observable
S
log
of
(iii).
Moreover,
the
telecore
structure
T
An
of
(ii),
the
contact
structure
H
An
of
(ii),
and
the
observable
S
log
of
(iii)
are
not
simultaneously
compatible
[but
cf.
Remark
3.7.3,
(ii),
below].
(v)
The
unique
vertex
of
the
second
row
of
D
is
a
nexus
of
Γ
D
.
More-
over,
D
is
totally
-rigid,
and
the
natural
action
of
Z
on
the
infinite
linear
oriented
graph
Γ
D
≤1
extends
to
an
action
of
Z
on
D
by
nexus-classes
of
self-
equivalences
of
D.
Finally,
the
self-equivalences
in
these
nexus-classes
are
com-
patible
with
the
families
of
homotopies
that
constitute
the
cores
and
observ-
able
of
(i),
(iii);
these
self-equivalences
also
extend
naturally
[cf.
the
technique
of
extension
applied
in
Definition
3.5,
(vi)]
to
the
diagram
of
categories
[cf.
Definition
3.5,
(iv),
(a)]
that
constitutes
the
telecore
of
(ii),
in
a
fashion
that
is
compatible
with
both
the
family
of
homotopies
that
constitutes
this
telecore
structure
[cf.
Definition
3.5,
(iv),
(b)]
and
the
contact
structure
H
An
of
(ii).
Proof.
In
the
following,
if
φ
is
a
functor
appearing
in
D,
then
let
us
write
[φ]
for
the
path
on
the
underlying
oriented
graph
Γ
D
of
D
determined
by
the
edge
corresponding
to
φ
[cf.
§0].
Now
assertion
(i)
is
immediate
from
the
definitions
and
the
fact
that
the
algorithms
of
Corollary
1.10
are
“group-theoretic”
in
the
sense
that
they
are
expressed
in
language
that
depends
only
on
the
profinite
group
given
as
“input
data”.
Next,
we
consider
assertion
(ii).
The
portion
of
assertion
(ii)
concerning
φ
An
and
T
An
is
immediate
from
the
definitions
and
the
“group-theoretic”
algorithms
of
Corollary
1.10
[cf.
also
Proposition
3.2,
(ii),
(iii)].
Thus,
it
suffices
to
show
the
existence
of
a
contact
structure
H
An
as
described.
To
this
end,
let
us
first
ob-
serve
that
the
isomorphism
of
log
with
the
identity
functor
[cf.
Definition
3.1,
(iv);
Proposition
3.2,
(v)]
is
compatible
[in
the
evident
sense]
with
the
natural
tranforma-
−1
−1
,
η
,
η
}
∈L,∈L
†
.
On
the
other
hand,
this
compatibility
implies
tions
{η
,
η
that
one
may,
in
effect,
“contract”
D
≤2
down
to
a
single
vertex
[equipped
with
the
TOPICS
IN
ABSOLUTE
ANABELIAN
GEOMETRY
III
81
category
X
]
and
the
various
paths
from
to
Anab
down
to
a
single
edge
—
i.e.,
that,
up
to
redundancies,
one
is,
in
effect,
dealing
with
a
diagram
of
categories
with
two
vertices
“X
”
and
“Anab”
joined
by
two
[oriented]
edges
φ
An
,
π
An
.
Now
the
existence
of
a
family
of
homotopies
that
contains
the
collection
of
natural
transfor-
−1
}
∈L,∈L
†
follows
immediately.
This
completes
the
proof
of
mations
{η
,
η
,
η
assertion
(ii).
Next,
we
consider
assertion
(iii).
Write
E
log
for
the
set
of
ordered
pairs
of
paths
on
Γ
D
≤3
[i.e.,
the
underlying
oriented
graph
of
D
≤3
]
consisting
of
pairs
of
paths
of
the
following
three
types:
(1)
([λ
×
]
◦
[id
]
◦
[log]
◦
[γ],
[λ
×pf
]
◦
[id
+1
]
◦
[γ]),
where
[γ]
is
a
path
on
D
≤3
whose
terminal
vertex
lies
in
the
first
row
of
D
≤3
;
(2)
([λ
×
]
◦
[γ],
[λ
×pf
]
◦
[γ]),
where
[γ]
is
a
path
on
D
≤3
whose
terminal
vertex
lies
in
the
second
row
of
D
≤3
;
(3)
([γ],
[γ]),
where
[γ]
is
a
path
on
D
≤3
whose
terminal
vertex
lies
in
the
third
row
of
D
≤3
.
Then
one
verifies
immediately
that
E
log
satisfies
the
conditions
(a),
(b),
(c),
(d),
(e)
given
in
§0
for
a
saturated
set.
Moreover,
the
natural
transformation(s)
ι
log,
(respectively,
ι
×
)
determine(s)
the
homotopies
for
pairs
of
paths
of
type
(1)
(re-
spectively,
(2)).
Thus,
we
obtain
an
observable
S
log
,
as
desired.
Moreover,
it
is
immediate
from
the
definitions
—
i.e.,
in
essence,
because
the
various
Galois
groups
that
appear
remain
“undisturbed”
by
the
various
manipulations
involving
arithmetic
data
that
arise
from
“ι
log,
”,
“ι
×
”
—
that
this
family
of
homotopies
is
compatible
with
the
families
of
homotopies
that
constitute
the
core
and
telecore
structures
of
(i),
(ii).
This
completes
the
proof
of
assertion
(iii).
Next,
we
consider
assertion
(iv).
Suppose
that
D
≤2
admits
a
structure
of
core
on
D
≤1
in
a
fashion
that
is
compatible
with
the
observable
S
log
of
(iii).
Then
this
core
structure
determines,
for
∈
L,
a
homotopy
ζ
0
for
the
pair
of
paths
([id
+1
],
[id
]
◦
[log]);
thus,
by
composing
the
result
ζ
0
of
applying
λ
×
to
ζ
0
with
the
homotopy
ζ
1
associated
[via
S
log
]
to
the
pair
of
paths
([λ
×
]◦[id
]◦[log],
[λ
×pf
]◦
[id
+1
])
[of
type
(1)],
we
obtain
a
natural
transformation
ζ
1
=
ζ
1
◦
ζ
0
:
λ
×
◦
id
+1
→
λ
×pf
◦
id
+1
—
which,
in
order
for
the
desired
compatibility
to
hold,
must
coincide
with
the
homotopy
ζ
2
associated
[via
S
log
]
to
the
pair
of
paths
([λ
×
]◦[id
+1
],
[λ
×pf
]◦[id
+1
])
[of
type
(2)].
On
the
other
hand,
by
writing
out
explicitly
the
meaning
of
such
an
equality
ζ
1
=
ζ
2
,
we
conclude
that
we
obtain
a
contradiction
to
Lemma
3.4.
This
completes
the
proof
of
the
first
incompatibility
of
assertion
(iv).
The
proof
of
the
second
incompatibility
of
assertion
(iv)
is
entirely
similar.
That
is
to
say,
if
we
compose
on
the
right
with
[φ
+1
]
the
various
paths
that
appeared
in
the
proof
of
the
first
incompatibility,
then
in
order
to
apply
the
argument
applied
in
the
proof
of
the
first
incompatibility,
it
suffices
to
relate
the
paths
[id
+1
]
◦
[φ
+1
];
[id
]
◦
[log]
◦
[φ
+1
]
82
SHINICHI
MOCHIZUKI
[a
task
that
was
achieved
in
the
proof
of
the
first
incompatibility
by
applying
the
core
structure
whose
existence
was
assumed
in
the
proof
of
the
first
incompatibility].
In
the
present
situation,
applying
the
homotopy
η
+1
of
the
contact
structure
H
An
yields
a
homotopy
[φ
]
[id
+1
]◦[φ
+1
];
on
the
other
hand,
we
obtain
a
homotopy
1
[φ
]
[id
]
◦
[φ
]
[id
]
◦
[β
]
◦
[log]
◦
[φ
+1
]
[id
]
◦
[log]
◦
[φ
+1
]
by
applying
the
homotopy
η
of
the
contact
structure
H
An
,
followed
by
the
homo-
topies
of
the
telecore
T
An
,
followed
by
the
homotopy
η
of
the
contact
structure
H
An
.
Thus,
by
applying
the
argument
applied
in
the
proof
of
the
first
incompatibility,
we
obtain
two
mutually
contradictory
homotopies
[λ
×
]◦[φ
]
[λ
×pf
]◦[id
+1
]◦[φ
+1
].
This
completes
the
proof
of
the
second
incompatibility
of
assertion
(iv).
Finally,
we
consider
assertion
(v).
The
total
-rigidity
in
question
follows
immediately
from
Proposition
3.2,
(iv)
[cf.
also
the
final
portion
of
Proposition
3.2,
(v)].
The
remainder
of
assertion
(v)
follows
immediately
from
the
definitions.
This
completes
the
proof
of
assertion
(v).
Remark
3.6.1.
(i)
The
“output”
of
the
“log-Frobenius
observable”
S
log
of
Corollary
3.6,
(iii),
may
be
summarized
intuitively
in
the
following
diagram:
...
Π
+1
∼
→
...
Π
+1
∼
→
×
k
+1
→
×
(k
+1
)
pf
∼
→
Π
Π
←
×
→
k
...
×
(k
)
pf
∼
→
Π
−1
∼
→
Π
−1
...
←
×
k
−1
...
×
→
(k
−1
)
pf
...
×
—
where
the
arrows
“→”
are
the
natural
morphisms
[cf.
ι
×
!];
k
,
for
∈
L,
is
a
×
copy
of
“k
”
that
arises,
via
id
,
from
the
vertex
of
D
≤1
;
the
arrows
“←”
are
the
×
inclusions
arising
from
the
fact
that
k
is
obtained
by
applying
the
log-Frobenius
×
functor
log
to
k
+1
[cf.
ι
log,
!];
the
isomorphic
“Π
’s”
that
act
on
the
various
×
k
’s
and
their
perfections
correspond
to
the
coricity
of
E
[cf.
Corollary
3.6,
(i)].
Finally,
the
incompatibility
assertions
of
Corollary
3.6,
(iv),
may
be
thought
of
as
a
statement
of
the
non-existence
of
some
“universal
reference
model”
×
k
model
×
that
maps
isomorphically
to
the
various
k
’s
in
a
fashion
that
is
compatible
with
the
various
arrows
“→”,
“←”
of
the
above
diagram
—
cf.
also
Corollary
3.7,
(iv),
below.
TOPICS
IN
ABSOLUTE
ANABELIAN
GEOMETRY
III
83
(ii)
In
words,
the
content
of
Corollary
3.6
may
be
summarized
as
follows
[cf.
the
“intuitive
diagram”
of
(i)]:
(a)
The
Galois
groups
that
act
on
the
various
objects
under
consideration
are
compatible
with
all
of
the
operations
involved
—
in
particular,
the
op-
erations
constituted
by
the
functors
log,
κ
An
,
φ
An
and
the
various
related
families
of
homotopies
—
cf.
the
coricity
asserted
in
Corollary
3.6,
(i).
(b)
By
contrast,
the
operation
constituted
by
the
log-Frobenius
functor
[as
“observed”
via
the
observable
S
log
]
is
not
compatible
with
the
field
struc-
ture
of
the
fields
[i.e.,
“k”]
involved
[cf.
Corollary
3.6,
(iv)].
(c)
As
a
consequence
of
(a),
the
“group-theoretic
reconstruction”
of
the
base
field
via
Corollary
1.10
is
compatible
with
all
of
the
operations
involved,
except
“momentarily”
when
log
acts
on
the
output
of
φ
An
—
an
operation
which
“temporarily
obliterates”
the
field
structure
of
this
output,
although
this
field
structure
may
be
recovered
by
projecting
back
down
to
E
[cf.
(a)]
and
applying
κ
An
.
This
sort
of
“conditional
compatibility”
—
i.e.,
up
to
a
“brief
temporary
exception”
—
is
expressed
in
the
telecoricity
asserted
in
Corollary
3.6,
(ii).
In
particular,
if
one
thinks
of
the
various
operations
involved
as
being
“software”
[cf.
Remark
1.9.8],
then
the
projection
to
E
—
i.e.,
the
operation
of
looking
at
the
Galois
group
—
is
compatible
with
simultaneous
execution
of
all
the
“software”
[in
particular,
including
log!]
under
consideration;
the
“group-theoreticity”
of
the
algorithms
of
Corollary
1.10
implies
that
Anab,
κ
An
satisfy
a
similar
“compatibility
with
simultaneous
execution
of
all
software”
[cf.
Remark
3.1.2]
property.
Remark
3.6.2.
(i)
The
reasoning
that
lies
behind
the
name
“log-Frobenius
functor”
may
be
understood
as
follows.
At
a
very
naive
level,
the
natural
logarithm
may
be
thought
of
as
a
sort
of
“raising
to
the
-th
power”
[where
→
0
is
some
indefinite
positive
infinitesimal]
—
i.e.,
“”
plays
the
role
in
characteristic
[
→]0
of
“p”
in
charac-
teristic
p
>
0.
More
generally,
the
logarithm
frequently
appears
in
the
context
of
Frobenius
actions,
in
particular
in
discussions
involving
canonical
coordinates,
such
as
in
[Mzk1],
Chapter
III,
§1.
(ii)
In
general,
Frobenius
morphisms
may
be
thought
of
as
“compression
mor-
phisms”.
For
instance,
this
phenomenon
may
be
seen
in
the
most
basic
example
of
a
Frobenius
morphism
in
characteristic
p
>
0,
i.e.,
the
morphism
t
→
t
p
on
F
p
[t].
Put
another
way,
the
“compression”
operation
inherent
in
a
Frobenius
morphism
may
be
thought
of
as
an
approximation
of
some
sort
of
“absolute
con-
stant
object”
[such
as
F
p
].
In
the
context
of
the
log-Frobenius
functor,
this
sort
of
compression
phenomenon
may
be
seen
in
the
pre-log-shells
defined
in
Definition
3.1,
(iv),
which
will
play
a
key
role
in
the
theory
of
§5
below.
84
SHINICHI
MOCHIZUKI
(iii)
Whereas
the
log-Frobenius
functor
obliterates
the
field
[or
ring]
structure
[cf.
Remark
3.6.1,
(ii),
(b)]
of
the
fields
involved,
the
usual
Frobenius
morphism
in
positive
characteristic
is
compatible
with
the
ring
structure
of
the
rings
involved.
On
the
other
hand,
unlike
generically
smooth
morphisms,
the
Frobenius
morphism
in
positive
characteristic
has
the
effect
of
“obliterating
the
differentials”
of
the
schemes
involved.
Remark
3.6.3.
The
diagram
D
of
Corollary
3.6
—
cf.,
especially,
the
first
two
rows
D
≤2
and
the
various
natural
actions
of
Z
discussed
in
Corollary
3.6,
(v)
—
may
be
thought
of
as
a
sort
of
combinatorial
version
of
G
m
—
cf.
the
point
of
view
of
Remark
1.9.7.
Remark
3.6.4.
One
verifies
immediately
that
one
may
give
a
tempered
version
of
Propositions
3.2,
3.3;
Corollary
3.6
[cf.
Remarks
1.9.1,
1.10.2].
Remark
3.6.5.
The
notions
of
“core”
and
“telecore”
are
reminiscent
of
certain
aspects
of
“Hensel’s
lemma”
[cf.,
e.g.,
[Mzk21],
Lemma
2.1].
That
is
to
say,
if
one
compares
the
successive
approximation
operation
applied
in
the
proof
of
Hensel’s
lemma
[cf.,
e.g.,
the
proof
of
[Mzk21],
Lemma
2.1]
to
the
various
operations
[in
the
form
of
functors]
that
appear
in
a
diagram
of
categories,
then
one
is
led
to
the
following
analogy:
cores
←→
sets
of
solutions
of
“étale”,
i.e.,
“slope
zero”
equations
telecores
←→
sets
of
solutions
of
“positive
slope”
equations
—
i.e.,
where
one
thinks
of
applications
of
Hensel’s
lemma
in
the
context
of
mixed
characteristic,
so
the
property
of
being
“étale
in
characteristic
p
>
0”
may
be
regarded
as
corresponding
to
“slope
zero”.
That
is
to
say,
the
“étale
case”
of
Hensel’s
lemma
is
the
easiest
to
understand.
In
this
“étale
case”,
the
invertibil-
ity
of
the
Jacobian
matrix
involved
implies
that
when
one
executes
each
successive
approximation
operation,
the
set
of
solutions
lifts
uniquely,
i.e.,
“transports
isomor-
phically”
through
the
operation.
This
sort
of
“isomorphic
transport”
is
reminiscent
of
the
definition
of
a
core
on
a
diagram
of
categories.
On
the
other
hand,
the
“pos-
itive
slope
case”
of
Hensel’s
lemma
is
a
bit
more
complicated
[cf.,
e.g.,
the
proof
of
[Mzk21],
Lemma
2.1].
That
is
to
say,
although
the
set
of
solutions
does
not
quite
“transport
isomorphically”
in
the
simplest
most
transparent
fashion,
the
fact
that
the
Jacobian
matrix
involved
is
invertible
up
to
a
factor
of
p
implies
that
the
set
of
solutions
“essentially
transports
isomorphically,
up
to
a
brief
temporary
lag”
—
cf.
the
“brief
temporary
exception”
of
Remark
3.6.1,
(ii),
(c).
Put
another
way,
if
one
thinks
in
terms
of
connections
on
bases
on
which
p
is
nilpotent,
in
which
case
formal
integration
takes
the
place
of
the
“successive
approximation
operation”
of
Hensel’s
lemma,
then
one
has
the
following
analogy:
cores
←→
vanishing
p
n
-curvature
telecores
←→
nilpotent
p
n
-curvature
[where
we
refer
to
[Mzk4],
Chapter
II,
§2;
[Mzk7],
§2.4,
for
more
on
“p
n
-curvature”]
—
cf.
Remark
3.7.2
below.
TOPICS
IN
ABSOLUTE
ANABELIAN
GEOMETRY
III
85
Remark
3.6.6.
(i)
In
the
context
of
the
analogy
between
telecores
and
“positive
slope
situa-
tions”
discussed
in
Remark
3.6.5,
one
question
that
may
occur
to
some
readers
is
the
following:
What
are
the
values
of
the
positive
slopes
implicitly
involved
in
a
telecore?
At
the
time
of
writing,
it
appears
to
the
author
that,
relative
to
this
analogy,
one
should
regard
telecores
as
containing
“all
positive
slopes”,
or,
alternatively,
“positive
slopes
of
an
indeterminate
nature”,
which
one
may
think
of
as
corresponding
to
the
lengths
of
the
various
paths
emanating
from
the
core
vertex
that
one
may
travel
along
before
descending
back
down
to
the
core
[cf.
Remark
3.5.1].
Indeed,
from
the
point
of
view
of
the
analogy
[discussed
in
Remark
3.7.2
below]
with
the
uniformizing
MF
∇
-objects
constructed
in
[Mzk1],
[Mzk4],
this
is
natural,
since
uniformizing
MF
∇
-objects
also
involve,
in
effect,
“all
positive
slopes”.
Moreover,
since
telecores
are
of
an
“abstract,
combinatorial
nature”
[cf.
Remark
1.9.7]
—
i.e.,
not
of
a
“linear,
module-theoretic
nature”,
as
is
the
case
with
MF
∇
-objects
—
it
seems
somewhat
natural
that
this
“combinatorial
non-linearity”
should
interfere
with
any
attempts
to
“separate
out
the
various
distinct
positive
slopes”,
via,
for
instance,
a
“linear
filtration”,
as
is
often
possible
in
the
case
of
MF
∇
-objects.
(ii)
From
the
point
of
view
of
the
analogy
with
[uniformizing]
MF
∇
-objects,
one
has
the
following
[rough]
correspondence:
slope
zero
←→
Frobenius
“”
(an
isomorphism)
positive
slope
←→
Frobenius
“”
p
n
·
(an
isomorphism)
[where
“”
is
to
be
understood
as
shorthand
for
the
phrase
“acts
via”;
n
is
a
positive
integer].
Perhaps
the
most
fundamental
example
in
the
p-adic
theory
of
such
a
[uniformizing]
MF
∇
-object
arises
from
the
p-adic
Galois
representation
obtained
by
extracting
p-power
roots
of
the
standard
unit
U
on
the
multiplicative
group
G
m
over
Z
p
,
in
which
case
the
“positive
slope”
involved
corresponds
to
the
action
dlog(U
)
=
dU/U
→
p
·
dlog(U
)
induced
by
the
Frobenius
morphism
U
→
U
p
.
In
the
situation
of
Corollary
3.6,
an
analogue
of
this
sort
of
correspondence
may
be
seen
in
the
“temporary
failure
of
coricity”
[cf.
Remark
3.6.1,
(ii),
(c);
the
failure
of
coricity
documented
in
Corollary
3.6,
(iv)]
of
the
“mono-anabelian
telecore”
of
Corollary
3.6,
(ii).
That
is
to
say,
mutiplication
by
a
positive
power
of
p
corresponds
precisely
to
this
“temporary
failure
of
coricity”,
a
failure
that
is
remedied
[where
the
“remedy”
corresponds
to
the
isomorphism
that
appears
by
“pealing
off
”
an
appropriate
power
of
p]
by
projecting
back
down
to
Anab,
an
operation
which
[in
light
of
the
“group-theoretic
nature”
of
the
algorithms
applied
in
κ
An
]
induces
an
isomorphism
of,
for
instance,
the
base-field
reconstructed
after
the
application
of
log
with
the
base-field
that
was
reconstructed
prior
to
the
application
of
log.
Remark
3.6.7.
Note
that
in
the
situation
of
Corollary
3.6
[cf.
also
the
ter-
minology
introduced
in
Definition
3.5],
although
we
have
formulated
things
in
the
86
SHINICHI
MOCHIZUKI
language
of
categories
and
functors,
in
fact,
the
mathematical
constructs
in
which
we
are
ultimately
interested
have
much
more
elaborate
structures
than
categories
and
functors.
That
is
to
say:
In
fact,
what
we
are
really
interested
in
is
not
so
much
“categories”
and
“functors”,
but
rather
“types
of
data”
and
“operations”
[i.e.,
algorithms!]
that
convert
some
“input
type
of
data”
into
some
“output
type
of
data”.
One
aspect
of
this
state
of
affairs
may
be
seen
in
the
fact
that
the
crucial
functors
log,
κ
An
of
Corollary
3.6
are
equivalences
of
categories
[which,
moreover,
are,
in
certain
cases,
isomorphic
to
the
identity
functor!
—
cf.
Definition
3.1,
(iv),
(vi)]
—
i.e.,
from
the
point
of
view
of
the
purely
category-theoretic
structure
[cf.,
e.g.,
the
point
of
view
of
[Mzk14],
[Mzk16],
[Mzk17]!]
of
“X
”,
“E”,
or
“Anab”,
these
functors
are
“not
doing
anything”.
On
the
other
hand,
from
the
point
of
view
of
“types
of
data”
and
“operations”
on
these
“types
of
data”
[cf.
Remark
3.6.1],
the
operations
constituted
by
the
functors
log,
κ
An
are
highly
nontrivial.
To
some
extent,
this
state
of
affairs
may
be
remedied
by
working
with
appropriate
observables
[i.e.,
which
serve
to
project
the
operations
constituted
by
functors
between
different
categories
down
into
arrows
in
a
single
category
—
cf.
Remark
3.5.1],
as
in
Corollary
3.6,
(iii),
(iv).
Nevertheless,
the
use
of
observables
does
not
constitute
a
fundamental
solution
to
the
issue
raised
above.
It
is
the
hope
of
the
author
to
remedy
this
state
of
affairs
in
a
more
definitive
fashion
in
a
future
paper
by
introducing
appropriate
“enhancements”
to
the
usual
theory
of
categories
and
functors.
To
understand
what
is
gained
in
Corollary
3.6
by
the
mono-anabelian
theory
of
§1,
it
is
useful
to
consider
the
following
“bi-anabelian
analogue”
of
Corollary
3.6.
Corollary
3.7.
(MLF-Galois-theoretic
Bi-anabelian
Log-Frobenius
In-
compatibility)
In
the
notation
and
conventions
of
Corollary
3.6,
suppose,
further,
that
T
=
TF.
Consider
the
diagram
of
categories
D
†
...
...
X
×
E
X
pr
+1
log
X
−→
X
×
E
X
⏐
⏐
pr
log
X
−→
X
×
E
X
...
pr
−1
...
X
⏐
⏐
⏐
⏐
λ
×
λ
×pf
N
⏐
⏐
E
—
where
the
second
to
fourth
rows
of
D
†
are
identical
to
the
second
to
fourth
rows
†
of
the
diagram
D
of
Corollary
3.6;
D
≤1
is
obtained
by
applying
the
“categorical
fiber
product”
(−)
×
E
X
[cf.
§0]
to
D
≤1
;
pr
denotes
the
projection
to
the
first
factor
on
the
copy
of
X
×
E
X
at
the
vertex
∈
L.
Also,
let
us
write
π
:
X
×
E
X
→
X
TOPICS
IN
ABSOLUTE
ANABELIAN
GEOMETRY
III
87
for
the
projection
to
the
second
factor
on
the
copy
of
X
×
E
X
at
the
vertex
∈
L,
D
‡
for
the
result
of
appending
these
arrows
π
to
D
†
—
where
we
think
of
the
codomain
“X
”
of
the
arrows
π
as
a
new
“core”
vertex
lying
in
the
first
row
‡
of
D
‡
“under”
the
various
copies
of
“X
×
E
X
”
at
the
vertices
of
L
—
and
D
≤n
†
[where
n
∈
{1,
2,
3,
4}]
for
the
subdiagram
of
D
‡
constituted
by
D
≤n
,
together
with
the
newly
appended
arrows
π
.
Then:
†
‡
†
(i)
D
†
=
D
≤4
(respectively,
D
≤1
)
admits
a
natural
structure
of
core
on
D
≤3
†
(respectively,
D
≤1
).
That
is
to
say,
loosely
speaking,
E
“forms
a
core”
of
the
func-
‡
†
tors
in
D
;
the
“second
factor”
X
“forms
a
core”
of
the
functors
in
D
≤1
.
[Thus,
we
think
of
the
second
factor
of
the
various
fiber
product
categories
X
×
E
X
as
being
a
“universal
reference
model”
—
cf.
Remark
3.7.3
below.]
(ii)
Write
δ
X
:
X
→
X
×
E
X
for
the
natural
diagonal
functor
and
∼
bi
θ
bi
:
pr
bi
1
→
pr
2
bi
for
the
isomorphism
between
the
two
projection
functors
pr
bi
1
,
pr
2
:
X
×
E
X
→
X
that
arises
from
the
functoriality
—
i.e.,
the
bi-anabelian
[or
“Grothendieck
Conjecture”-type]
portion
[cf.
Remark
1.9.8]
—
of
the
“group-theoretic”
algorithms
of
Corollary
1.10.
Then
δ
X
is
an
equivalence
of
categories,
a
quasi-inverse
for
which
is
given
by
the
projection
to
the
second
factor
π
X
:
X
×
E
X
→
X
;
θ
bi
∼
determines
an
isomorphism
θ
X
:
δ
X
◦
π
X
→
id
X
×
E
X
.
Moreover,
δ
X
gives
rise
to
a
†
telecore
structure
T
δ
on
D
≤1
,
whose
underlying
diagram
of
categories
we
denote
‡
‡
by
D
δ
,
by
appending
to
D
≤1
telecore
edges
X
⏐
⏐
δ
...
δ
+1
...
X
×
E
X
log
X
−→
X
×
E
X
δ
−1
log
X
−→
X
×
E
X
...
...
†
given
by
copies
of
δ
X
,
from
the
core
X
to
the
various
copies
of
X
×
E
X
in
D
≤1
which
we
denote
by
δ
,
for
∈
L.
That
is
to
say,
loosely
speaking,
δ
X
determines
†
a
telecore
structure
on
D
≤1
.
Finally,
let
us
write
D
∗
for
the
diagram
of
categories
‡
obtained
by
gluing
[in
the
evident
sense]
D
δ
‡
to
D
‡
along
D
≤1
and
then
appending
an
edge
δ
X
−→
X
from
the
core
vertex
of
D
δ
‡
to
the
vertex
at
[i.e.,
the
unique
vertex
of
the
second
row
of
D
‡
]
given
by
a
copy
of
the
identity
functor;
for
each
∈
L,
let
us
write
0
1
[γ
]
for
the
path
on
Γ
D
∗
of
length
0
at
and
[γ
]
for
the
path
on
Γ
D
∗
of
length
88
SHINICHI
MOCHIZUKI
2
that
starts
from
,
descends
via
π
to
the
core
vertex,
and
returns
to
via
the
telecore
edge
δ
.
Then
the
collection
of
natural
transformations
−1
−1
,
θ
,
θ
}
∈L
{θ
,
θ
—
where
we
write
θ
for
the
identity
natural
transformation
from
the
arrow
δ
:
X
→
X
to
the
composite
arrow
pr
◦
δ
:
X
→
X
and
∼
∗
∗
θ
:
D
[γ
1
]
→
D
[γ
0
]
for
the
isomorphism
arising
from
θ
X
—
generate
a
family
of
homotopies
H
δ
on
D
∗
[hence,
in
particular,
by
restriction,
a
contact
structure
on
the
telecore
∗
∗
admits
a
natural
structure
of
core
on
D
≤3
in
a
fashion
T
δ
].
Finally,
D
∗
=
D
≤4
†
†
compatible
with
the
core
structure
of
D
≤4
on
D
≤3
discussed
in
(i)
[that
is
to
say,
loosely
speaking,
E
“forms
a
core”
of
the
functors
in
D
∗
].
(iii)
Write
ι
log,
:
λ
×
◦
pr
◦
log
X
→
λ
×pf
◦
pr
+1
,
ι
×
=
ι
×
:
λ
×
→
λ
×pf
def
for
the
natural
transformations
determined
by
the
natural
transformations
of
Corol-
lary
3.6,
(iii).
Then
these
natural
transformations
ι
log,
,
ι
×
belong
to
a
family
of
†
†
that
determines
on
D
≤3
a
structure
of
observable
S
†
log
on
homotopies
on
D
≤3
†
D
≤2
and,
moreover,
is
compatible
with
the
families
of
homotopies
that
constitute
the
core
and
telecore
structures
of
(i),
(ii).
†
(iv)
The
diagram
of
categories
D
≤2
does
not
admit
a
structure
of
core
on
†
D
≤1
which
[i.e.,
whose
constituent
family
of
homotopies]
is
compatible
with
[the
constituent
family
of
homotopies
of
]
the
observable
S
†
log
of
(iii).
Moreover,
the
telecore
structure
T
δ
of
(ii),
the
family
of
homotopies
H
δ
of
(ii),
and
the
observable
S
†
log
of
(iii)
are
not
simultaneously
compatible.
(v)
The
vertex
of
the
second
row
of
D
∗
is
a
nexus
of
Γ
D
∗
.
Moreover,
D
∗
is
totally
-rigid,
and
the
natural
action
of
Z
on
the
infinite
linear
oriented
graph
Γ
†
extends
to
an
action
of
Z
on
D
∗
by
nexus-classes
of
self-equivalences
of
D
≤1
D
∗
.
Finally,
the
self-equivalences
in
these
nexus-classes
are
compatible
with
H
δ
[cf.
(ii)],
as
well
as
with
the
families
of
homotopies
that
constitute
the
cores,
telecore,
and
observable
of
(i),
(ii),
(iii).
Proof.
The
proofs
of
the
various
assertions
of
the
present
Corollary
3.7
are
entirely
similar
to
the
proofs
of
the
corresponding
assertions
of
Corollary
3.6.
Remark
3.7.1.
In
some
sense,
the
purpose
of
Corollary
3.7
is
to
examine
what
happens
if
the
mono-anabelian
theory
of
§1
is
not
available,
i.e.,
if
one
is
in
a
situation
in
which
one
may
only
apply
the
bi-anabelian
version
of
this
theory.
This
is
the
main
reason
for
our
assumption
that
“T
=
TF”
in
Corollary
3.7
—
that
is
TOPICS
IN
ABSOLUTE
ANABELIAN
GEOMETRY
III
89
to
say,
when
T
=
TM,
one
is
obliged
to
apply
Proposition
3.2,
(v),
a
result
whose
proof
requires
one
to
invoke
the
mono-anabelian
theory
of
§1.
Remark
3.7.2.
The
“Frobenius-theoretic”
point
of
view
of
Remarks
3.6.2,
3.6.5,
3.6.6
motivates
the
following
observation:
The
situation
under
consideration
in
Corollaries
3.6,
3.7
is
structurally
reminiscent
of
the
situation
encountered
in
the
p-adic
crystalline
theory,
for
instance,
when
one
considers
the
MF
∇
-objects
of
[Falt].
That
is
to
say,
the
core
E
plays
the
role
of
the
“absolute
constants”,
given,
for
instance,
in
the
p-adic
theory
by
[absolutely]
unramified
extensions
of
Z
p
.
The
isomorphism
∼
bi
θ
bi
:
pr
bi
1
→
pr
2
bi
[cf.
Corollary
3.7,
(ii)]
between
the
two
projection
functors
pr
bi
1
,
pr
2
:
X
×
E
X
→
X
is
formally
reminiscent
of
the
notion
of
a(n)
[integrable]
connection
in
the
crystalline
theory.
The
diagonal
functor
δ
X
:
X
→
X
×
E
X
[cf.
Corollary
3.7,
(ii)]
is
formally
reminiscent
of
the
diagonal
embedding
into
the
divided
power
envelope
of
the
product
of
a
scheme
with
itself
in
the
crystalline
theory.
Moreover,
since,
in
the
crystalline
theory,
this
divided
power
envelope
may
itself
be
regarded
as
a
crystal,
the
[various
divided
powers
of
the
ideal
defining
the]
diagonal
embedding
may
then
be
regarded
as
a
sort
of
Hodge
filtration
on
this
crys-
tal.
That
is
to
say,
the
telecore
structure
of
Corollary
3.7,
(ii),
may
be
regarded
as
corresponding
to
the
Hodge
filtration,
or,
for
instance,
in
the
context
of
the
theory
of
indigenous
bundles
[cf.,
e.g.,
[Mzk1],
[Mzk4]],
to
the
Hodge
section.
Thus,
from
this
point
of
view,
the
second
incompatibility
of
assertion
(iv)
of
Corollaries
3.6,
3.7,
is
reminiscent
of
the
fact
that
[in
general]
the
Frobenius
action
on
the
crystal
underlying
an
MF
∇
-object
fails
to
preserve
the
Hodge
filtration.
For
instance,
in
the
case
of
indigenous
bundles,
this
failure
to
preserve
the
Hodge
section
may
be
regarded
as
a
consequence
of
the
isomorphicity
of
the
Kodaira-Spencer
morphism
associated
to
the
Hodge
section.
On
the
other
hand,
the
log-Frobenius-compatibility
of
the
mono-anabelian
models
discussed
in
Corollary
3.6
may
be
regarded
as
corre-
sponding
to
canonical
Frobenius
actions
on
the
crystals
constituted
by
the
divided
power
envelopes
discussed
above
—
cf.
the
uniformizing
MF
∇
-objects
constructed
in
[Mzk1],
[Mzk4].
Moreover,
the
compatibility
of
the
coricity
of
E,
Anab
with
the
telecore
and
contact
structures
of
Corollary
3.6,
(ii),
on
the
one
hand,
and
the
“log-
Frobenius
observable”
S
log
of
Corollary
3.6,
(iii),
on
the
other
hand,
is
reminiscent
of
the
construction
of
the
Galois
representation
associated
to
an
MF
∇
-object
by
considering
the
submodule
that
lies
in
the
0-th
step
of
the
Hodge
filtration
and,
moreover,
is
fixed
by
the
action
of
Frobenius
[cf.
Remark
3.7.3,
(ii),
below].
Thus,
in
summary,
we
have
the
following
“dictionary”:
the
coricity
of
E
←→
absolutely
unramified
constants
bi-anabelian
isomorphism
of
projection
functors
←→
integrable
connections
diagonal
functor
δ
X
telecore
str.
←→
Hodge
filtration/section
90
SHINICHI
MOCHIZUKI
bi-anabelian
log-incompatibility
←→
Kodaira-Spencer
isomorphism
“forgetful”
functor
φ
An
telecore
str.
←→
underlying
vector
bundle
of
MF
∇
-object
mono-anabelian
log-compatibility
←→
[positive
slope!]
uniformizing
MF
∇
-objects
[where
“str.”
stands
for
“structure”].
This
analogy
with
MF
∇
-objects
will
be
developed
further
in
§5
below.
Remark
3.7.3.
The
significance
of
Corollary
3.7
in
the
context
of
our
discussion
of
the
mono-anabelian
versus
the
bi-anabelian
approach
to
anabelian
geometry
[cf.
Remark
1.9.8]
may
be
understood
as
follows.
(i)
We
begin
by
considering
the
conditions
that
we
wish
to
impose
on
the
framework
in
which
we
are
to
work.
First
of
all,
we
wish
to
have
some
sort
of
fixed
reference
model
of
“X
”.
The
fact
that
this
model
is
to
be
fixed
throughout
the
discussion
then
translates
into
the
requirement
that
this
copy
of
X
be
a
core,
relative
to
the
various
“operations”
performed
during
the
discussion.
On
the
other
hand,
one
does
not
wish
for
this
model
to
remain
“completely
unrelated
to
the
operations
of
interest”,
but
rather
that
it
may
be
related,
or
compared,
to
the
various
copies
of
this
model
that
appear
as
one
executes
the
operations
of
interest.
In
our
situation,
we
wish
to
be
able
to
relate
the
“fixed
reference
model”
to
the
copies
of
this
model
—
i.e.,
“log-subject
copies”
—
that
are
subject
to
the
log-Frobenius
operation
[i.e.,
functor
—
cf.
Remark
3.6.7].
Moreover,
since
the
log-Frobenius
functor
is
isomorphic
to
the
identity
functor
[cf.
Proposition
3.2,
(v)],
hence
may
only
be
“properly
understood”
in
the
context
of
the
natural
transformations
“ι
×
”
and
“ι
log
”,
we
wish
for
everything
that
we
do
to
be
compatible
with
the
operation
of
“making
an
observation”
via
these
natural
transformations.
Thus,
in
summary,
the
main
conditions
that
we
wish
to
impose
on
the
framework
in
which
we
are
to
work
are
the
following:
(a)
coricity
of
the
model;
(b)
comparability
of
the
model
to
log-subject
copies
of
the
model;
(c)
consistent
observability
of
the
various
operations
executed
[especially
log].
In
the
context
of
the
various
assertions
of
Corollaries
3.6,
3.7,
these
three
aspects
correspond
as
follows:
(a)
←→
the
coricity
of
(i),
the
“coricity
portion”
of
the
telecore
structure
of
(ii),
(b)
←→
the
telecore
and
contact
structures/families
of
homotopies
of
(ii),
(c)
←→
the
“log-observable”
of
(iii).
[Here,
we
refer
to
the
content
of
Definition
3.5,
(iv),
(b),
as
the
“coricity
portion”
of
a
telecore
structure.]
In
the
case
of
Corollary
3.7,
the
“fixed
reference
model”
is
realized
by
applying
a
“category-theoretic
base-change”
(−)
×
E
X
,
as
in
Corollary
3.7,
i.e.,
the
copy
of
“X
”
used
to
effect
this
base-change
serves
as
the
“fixed
reference
model”;
in
the
case
of
Corollary
3.6,
the
“fixed
reference
model”
is
given
by
“Anab”
[i.e.,
especially,
the
second
piece
of
parenthesized
data
“(−,
−)”
in
the
definition
of
Anab
—
cf.
Definition
3.1,
(vi)].
Also,
we
observe
that
the
second
incompatibility
of
TOPICS
IN
ABSOLUTE
ANABELIAN
GEOMETRY
III
91
assertion
(iv)
of
Corollaries
3.6,
3.7
asserts,
in
effect,
that
neither
of
the
approaches
of
these
two
corollaries
succeeds
in
simultaneously
realizing
conditions
(a),
(b),
(c),
in
the
strict
sense.
(ii)
Let
us
take
a
closer
look
at
the
mono-anabelian
approach
of
Corollary
3.6
from
the
point
of
view
of
the
discussion
of
(i).
From
the
point
of
view
of
“operations
performed”,
this
approach
may
be
summarized
as
follows:
One
starts
with
“Π”,
applies
the
mono-anabelian
algorithms
of
Corollary
1.10
to
obtain
an
object
of
Anab,
then
forgets
the
“group-theoretic
origins”
of
such
objects
to
obtain
an
object
of
X
[cf.
the
telecore
structure
of
Corollary
3.6,
(ii)],
which
is
subject
to
the
action
of
log;
this
action
of
log
obliterates
the
ring
structure
[indeed,
it
obliterates
both
the
additive
and
multiplicative
structures!]
of
the
arithmetic
data
involved,
hence
leaving
behind,
as
an
invariant
of
log,
only
the
original
“Π”,
to
which
one
may
again
apply
the
mono-anabelian
algorithms
of
Corollary
1.10.
⎞
⎛
Π
⎞
⎟
⎜
⎜
⎟
⎠
⎝
⎜
⎜
⎝
⎟
⎟
⎠
⎛
Π
Π
×
k
An
×
k
⎛
Π
log
Π
⎞
⎜
⎟
⎟
⎜
⎝
⎠
×
k
An
The
point
of
the
mono-anabelian
approach
is
that
although
log
obliterates
the
ring
structures
involved
[cf.
the
second
incompatibility
of
Corollary
3.6,
(iv)],
E
—
i.e.,
“Π”
—
remains
constant
[up
to
isomorphism]
throughout
the
application
of
the
various
operations;
this
implies
that
the
“purely
group-theoretic
constructions”
of
Corollary
1.10
—
i.e.,
Anab,
κ
An
—
also
remain
constant
throughout
the
application
of
the
various
operations.
In
particular,
in
the
above
diagram,
despite
the
fact
×
that
log
obliterates
the
ring
structure
of
“k
”,
the
operations
executed
induce
an
isomorphism
between
all
the
“Π’s”
that
appear,
hence
an
isomorphism
between
the
×
initial
and
final
“(Π
k
An
)’s”.
At
a
more
technical
level,
this
state
of
affairs
may
be
witnessed
in
the
fact
that
although
[cf.
the
proof
of
the
second
incompatibility
of
Corollary
3.6,
(iv)]
there
exist
incompatible
composites
of
homotopies
involving
the
families
of
homotopies
that
constitute
the
telecore,
contact,
and
observable
structures
involved,
these
composites
become
compatible
as
soon
as
one
augments
the
various
paths
involved
with
a
path
back
down
to
the
core
vertex
“Anab”.
At
a
more
philosophical
level:
This
state
of
affairs,
in
which
the
application
of
log
does
not
immediately
×
yield
an
isomorphism
of
“k
’s”,
but
does
after
“pealing
off
the
operation
×
of
forgetting
the
group-theoretic
construction
of
k
An
”,
is
reminiscent
of
the
situation
discussed
in
Remark
3.6.6,
(ii),
concerning
“Frobenius
p
n
·
(an
isomorphism)”
[i.e.,
where
Frobenius
induces
an
isomorphism
after
“pealing
off
”
an
ap-
propriate
power
of
p].
(iii)
By
contrast,
the
bi-anabelian
approach
of
Corollary
3.7
may
be
understood
in
the
context
of
the
present
discussion
as
follows:
One
starts
with
an
arbitrarily
92
SHINICHI
MOCHIZUKI
×
declared
“model”
copy
“Π
k
model
”
of
X
,
then
forgets
the
fact
that
this
copy
was
arbitrarily
declared
a
model
[cf.
the
telecore
structure
of
Corollary
3.7,
(ii)];
this
×
yields
a
copy
“Π
k
”
of
X
on
which
log
acts
in
a
fashion
that
obliterates
the
ring
structure
of
the
arithmetic
data
involved,
hence
leaving
behind,
as
an
invariant
of
log,
only
the
original
“Π”.
⎛
Π
⎞
⎛
Π
⎞
⎜
⎜
⎝
⎟
⎟
⎠
⎜
⎜
⎝
⎟
⎟
⎠
×
k
model
×
k
Π
log
Thus,
unlike
the
case
with
the
mono-anabelian
approach,
if
one
tries
to
work
with
×
×
another
model
“Π
k
model
”
after
applying
log,
then
the
“k
model
”
portion
of
this
×
new
model
cannot
be
related
to
the
“k
model
”
portion
of
the
original
model
in
a
con-
sistent
fashion
—
i.e.,
such
a
relation
is
obstructed
by
log,
which
obliterates
the
ring
×
structure
of
k
model
.
Moreover,
unlike
the
case
with
the
mono-anabelian
approach,
there
is
“no
escape
route”
in
the
bi-anabelian
approach
[i.e.,
which
requires
the
use
of
models]
from
this
situation
given
by
taking
a
path
back
down
to
some
core
vertex
[i.e.,
such
as
“Anab”].
Relative
to
the
analogy
with
usual
Frobenius
actions
[cf.
Remark
3.6.6,
(ii)],
this
situation
is
reminiscent
of
the
Frobenius
action
on
the
ideal
defining
the
diagonal
of
a
divided
power
envelope
I⊆O
PD
S
×
S
[where
S
is,
say,
smooth
over
F
p
]
—
i.e.,
Frobenius
simply
maps
I
to
0
in
a
fashion
that
does
not
allow
one
to
“recover,
in
an
isomorphic
fashion,
by
pealing
off
a
power
of
p”.
[Indeed,
the
data
necessary
to
“peal
off
a
power
of
p”
consists,
in
essence,
of
a
Frobenius
lifting
—
which
is,
in
essence,
equivalent
to
the
datum
of
a
uniformizing
MF
∇
-object
—
cf.
Remark
3.7.2;
the
theory
of
[Mzk1],
[Mzk4].]
In
particular:
Although
it
is
difficult
to
give
a
completely
rigorous
formulation
of
the
?
question
“bi-anabelian
=⇒
mono-anabelian”
raised
in
Remark
1.9.8,
the
state
of
affairs
discussed
above
strongly
suggests
a
negative
answer
to
this
question.
(iv)
The
following
questions
constitute
a
useful
guide
to
understanding
better
the
gap
that
lies
between
the
“success
of
the
mono-anabelian
approach”
and
the
“failure
of
the
bi-anabelian
approach”,
as
documented
in
(i),
(ii),
(iii):
(a)
In
what
capacity
—
i.e.,
as
what
type
of
mathematical
object
[cf.
Re-
mark
3.6.7]
—
does
one
transport
—
i.e.,
“effect
the
coricity
of”
[cf.
con-
×
dition
(a)
of
(i)]
—
the
fixed
reference
model
of
“k
”
down
to
“future
log-generations”
[i.e.,
smaller
elements
of
L]?
(b)
On
precisely
what
type
of
data
[cf.
Remark
3.6.7]
does
the
comparison
[cf.
condition
(b)
of
(i)]
via
telecore/contact
structures
depend?
That
is
to
say,
in
the
mono-anabelian
approach,
the
answer
to
both
questions
is
given
by
E
[i.e.,
“Π”],
Anab;
by
contrast,
in
the
bi-anabelian
approach,
the
answer
TOPICS
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ANABELIAN
GEOMETRY
III
93
×
to
(b)
necessarily
requires
the
inclusion
of
the
“model
k
model
”
—
a
requirement
that
is
incompatible
with
the
coricity
required
by
(a)
[i.e.,
since
log
obliterates
the
ring
×
structure
of
k
model
].
(
Galois
[core]
)
Galois
+
log-subject
(
Galois
[of
model]
)
arithmetic
data
Fig.
1:
Mono-anabelian
comparison
only
requires
“Galois
input
data”.
(
Galois
[core]
)
Galois
+
model
arithmetic
data
?
∼
...
→
...
Galois
+
log-subject
arithmetic
data
Fig.
2:
Bi-anabelian
comparison
requires
“arithmetic
input
data”.
One
way
to
understand
this
state
of
affairs
is
as
follows.
If
one
attempts
to
construct
a
“bi-anabelian
version
of
Anab”,
then
the
requirement
of
coricity
means
that
the
×
“model
k
model
”
employed
in
the
bi-anabelian
reconstruction
algorithm
of
such
a
“bi-anabelian
version
of
Anab”
must
be
compatible
with
the
various
isomorphisms
×
×
∼
k
model
→
k
of
Remark
3.6.1,
(i)
—
where
we
recall
that
the
various
distinct
×
k
’s
are
related
to
one
another
by
log
—
i.e.,
compatible
with
the
“building”,
or
×
“edifice”,
of
k
’s
constituted
by
these
isomorphisms
together
with
the
diagram
of
Remark
3.6.1,
(i).
That
is
to
say,
in
order
for
the
required
coricity
to
hold,
this
bi-anabelian
reconstruction
algorithm
must
be
such
that
it
only
depends
on
×
the
ring
structure
of
k
model
“up
to
log”
—
i.e.,
the
algorithm
must
be
immune
to
the
confusion
[arising
from
log]
of
the
additive
and
multiplicative
structures
that
constitute
this
ring
structure.
On
the
other
hand,
the
bi-anabelian
approach
to
reconstruction
clearly
does
not
satisfy
this
property
[i.e.,
it
requires
that
the
ring
×
structure
of
k
model
be
left
intact].
Remark
3.7.4.
In
the
context
of
the
issue
of
distinguishing
between
the
mono-
anabelian
and
bi-anabelian
approaches
to
anabelian
geometry
[cf.
Remark
3.7.3],
one
question
that
is
often
posed
is
the
following:
Why
can’t
one
somehow
sneak
a
“fixed
refence
model”
into
a
“mono-
anabelian
reconstruction
algorithm”
by
finding,
for
instance,
some
copy
of
Q
or
Q
p
inside
the
Galois
group
“Π”
and
then
building
up
some
copy
of
the
hyper-
bolic
orbicurve
under
consideration
over
this
base
field
[i.e.,
this
copy
of
Q,
94
SHINICHI
MOCHIZUKI
Q
p
],
which
one
then
takes
as
one’s
“model”,
thus
allowing
one
to
“reduce”
mono-anabelian
problems
to
bi-anabelian
ones
[cf.
Remark
1.9.8]?
One
important
observation,
relative
to
this
question,
is
that
although
it
is
not
so
difficult
to
“construct”
such
copies
of
Q
or
Q
p
from
Π,
it
is
substantially
more
difficult
to
construct
copies
of
the
algebraic
closures
of
Q
or
Q
p
in
such
a
way
that
the
resulting
absolute
Galois
groups
are
isomorphic
to
the
appropriate
quotient
of
the
given
Galois
group
“Π”
in
a
functorial
fashion
[cf.
Remark
3.7.5
below].
Moreover,
once
one
constructs,
for
instance,
a
universal
pro-finite
étale
covering
of
an
appropriate
hyperbolic
orbicurve
on
which
Π
acts
[in
a
“natural”,
functorial
fashion],
one
must
specify
[cf.
question
(a)
of
Remark
3.7.3,
(iv)]
in
what
capacity
—
i.e.,
as
what
type
of
mathematical
object
—
one
transports
[i.e.,
“effects
the
coricity
of”]
this
pro-hyperbolic
orbicurve
model
down
to
“future
log-generations”.
Then
if
one
only
takes
a
naive
approach
to
these
issues,
one
is
ultimately
led
to
the
arbitrary
introduction
of
“models”
that
fail
to
be
immune
to
the
application
of
the
log-Frobenius
functor
—
that
is
to
say,
one
finds
oneself,
in
effect,
in
the
situation
of
the
“bi-anabelian
approach”
discussed
in
Remark
3.7.3.
Thus,
the
above
discussion
may
be
summarized
in
flowchart
form,
as
follows:
construction
of
model
[universal
pro-covering]
schemes
without
essential
use
of
Π
⇓
natural
functorial
action
of
Π
on
model
scheme
is
trivial
⇓
∼
must
supplement
model
scheme
with
Π
→
Gal(model
scheme)
⇓
essentially
equivalent
situation
to
“bi-anabelian
approach”.
Put
another
way,
if
one
tries
to
sneak
a
“fixed
refence
model”
that
may
be
con-
structed
without
essential
use
of
“Π”
into
a
“mono-anabelian
reconstruction
algo-
rithm”,
then
one
finds
oneself
confronted
with
the
following
two
mutually
exclusive
choices
concerning
the
type
of
mathematical
object
[cf.
question
(a)
of
Remark
3.7.3,
(iv)]
that
one
is
to
assign
to
this
model:
(∗)
the
model
arises
from
“Π”
=⇒
“functorially
trivial
model”;
(∗∗)
the
model
does
not
arise
from
“Π”
=⇒
“bi-anabelian
approach”.
In
particular,
Figures
1
and
2
of
Remark
3.7.3,
(iv),
are
not
[at
least
in
an
“a
priori
sense”]
“essentially
equivalent”.
Remark
3.7.5.
(i)
From
the
point
of
view
of
“constructing
models
of
the
base
field
from
Π”
[cf.
the
discussion
of
Remark
3.7.4],
one
natural
approach
to
the
issue
of
finding
TOPICS
IN
ABSOLUTE
ANABELIAN
GEOMETRY
III
95
“Galois-compatible
models”
is
to
work
with
Kummer
classes
of
scheme-theoretic
functions,
since
Kummer
classes
are
tautologically
compatible
with
Galois
actions.
[Indeed,
the
use
of
Kummer
classes
is
one
important
aspect
of
the
theory
of
§1.]
Moreover,
in
addition
to
being
“tautologically
Galois-compatible”,
Kummer
classes
also
have
the
virtue
of
fitting
into
a
container
def
(Π))
H(Π)
=
H
1
(Π,
μ
Z
[cf.
Corollary
1.10,
(d),
where
we
take
“Π
X
”
to
be
Π]
which
inherits
the
coricity
of
Π
[cf.
question
(a)
of
Remark
3.7.3,
(iv)]
in
a
very
natural,
tautological
fashion.
Thus,
once
one
characterizes,
in
a
“group-theoretic”
fashion,
the
“Kummer
subset”
of
this
container
H(Π)
[i.e.,
the
subset
constituted
by
the
Kummer
classes
that
arise
from
scheme-theoretic
functions],
it
remains
to
reconstruct
the
additive
structure
on
[the
union
with
{0}
of]
the
set
of
Kummer
classes
[cf.
the
theory
of
§1].
If
one
takes
the
point
of
view
of
the
question
posed
in
Remark
3.7.4,
then
it
is
tempting
to
try
to
use
“models”
solely
as
a
means
to
reconstruct
this
additive
structure.
This
approach,
which
combines
the
“purely
group-theoretic”
[i.e.,
“mono-
anabelian”]
container
H(Π)
with
the
indirect
use
of
“models”
to
recon-
struct
the
additive
structure
[or
the
Kummer
subset],
may
be
thought
of
as
a
sort
of
intermediate
alternative
between
the
“mono-anabelian”
and
“bi-anabelian”
approaches
discussed
so
far;
in
the
discussion
to
follow,
we
shall
refer
to
this
sort
of
intermediate
approach
as
“pseudo-mono-
anabelian”.
With
regard
to
implementing
this
pseudo-mono-anabelian
approach,
we
observe
that
the
“automorphism
version
of
the
Grothendieck
Conjecture”
[i.e.,
the
functo-
riality
of
the
algorithms
of
Corollary
1.10,
applied
to
automorphisms]
allows
one
to
conclude
that
the
additive
structure
“pulled
back
from
a
model
scheme
via
the
Kummer
map”
is
rigid
[i.e.,
remains
unaffected
by
automorphisms
of
Π].
On
the
other
hand,
the
“isomorphism
version
of
the
Grothendieck
Conjecture”
[i.e.,
the
functoriality
of
the
algorithms
of
Corollary
1.10,
applied
to
isomorphisms
—
cf.
the
isomorphism
θ
bi
of
Corollary
3.7,
(ii)]
allows
one
to
conclude
that
this
additive
structure
is
independent
of
the
choice
of
model.
(ii)
The
pseudo-mono-anabelian
approach
gives
rise
to
a
theory
that
satisfies
many
of
the
useful
properties
satisfied
by
the
mono-anabelian
theory.
Thus,
at
first
glance,
it
is
tempting
to
consider
simply
abandoning
the
mono-anabelian
approach,
in
favor
of
the
pseudo-mono-anabelian
approach.
Closer
inspection
reveals,
how-
ever,
that
the
situation
is
not
so
simple.
Indeed,
relative
to
the
coricity
requirement
of
Remark
3.7.3,
(i),
(a),
there
is
no
problem
with
allowing
the
“hidden
models”
on
which
the
pseudo-mono-anabelian
approach
depends
in
an
essential
way
to
remain
hidden.
On
the
other
hand,
the
issue
of
relating
[cf.
Remark
3.7.3,
(i),
(b)]
these
hidden
models
to
log-subject
copies
of
these
models
is
more
complicated.
Here,
the
central
problem
may
be
summarized
as
follows
[cf.
Remark
3.7.3,
(iv),
(a),
(b)]:
Problem
(∗
type
):
Find
a
type
of
mathematical
object
that
[in
the
context
of
the
framework
discussed
in
Remark
3.7.3,
(i)]
serves
as
a
common
type
of
mathematical
object
for
both
“coric
models”
and
“log-subject
copies”,
96
SHINICHI
MOCHIZUKI
thus
rendering
possible
the
comparison
of
“coric
models”
and
“log-subject
copies”.
That
is
to
say,
in
the
mono-anabelian
approach,
this
“common
type”
is
furnished
by
the
objects
that
constitute
E
and
[in
light
of
the
“group-theoreticity”
of
the
algo-
rithms
of
Corollary
1.10]
Anab;
in
the
bi-anabelian
approach,
the
“common
object”
is
furnished
by
the
“copy
of
X
that
appears
in
the
base-change
(−)
×
E
X
”.
Note
that
it
is
precisely
the
existence
of
this
“common
type
of
mathematical
object”
that
renders
possible
the
definition
of
the
telecore
structure
—
cf.,
especially,
the
functor
δ
X
of
Corollary
3.7,
(ii).
In
particular,
we
note
that
the
definition
of
the
diagonal
functor
δ
X
is
possible
precisely
because
of
the
equality
of
the
types
of
mathematical
object
involved
in
the
two
factors
of
X
×
E
X
.
On
the
other
hand,
if,
in
imple-
menting
the
pseudo-mono-anabelian
approach,
one
tries
to
use,
for
instance,
E
[i.e.,
without
including
the
“hidden
model”!],
then
although
this
yields
a
framework
in
which
it
is
possible
to
work
with
the
“mono-anabelian
container
H(Π)”,
this
does
not
allow
one
to
describe
the
contents
[i.e.,
the
Kummer
subset
with
its
ring
struc-
ture]
of
this
container.
That
is
to
say,
if
one
describes
these
“contents”
via
“hidden
models”,
then
the
data
contained
in
the
“common
type”
is
not
sufficient
for
the
operation
of
relating
this
description
to
the
“conventional
description
of
contents”
that
one
wishes
to
apply
to
the
log-subject
copies.
Indeed,
if
the
coric
models
and
log-subject
copies
only
share
the
container
H(Π),
but
not
the
description
of
its
contents
—
i.e.,
the
description
for
the
coric
models
is
some
“mysterious
descrip-
tion
involving
hidden
models”,
while
the
description
for
the
log-subject
copies
is
the
“standard
Kummer
map
description”
—
then,
a
priori,
there
is
no
reason
that
these
two
descriptions
should
coincide.
For
instance,
if
the
“mysterious
description”
is
not
related
to
the
“standard
description”
via
some
common
description
applied
to
a
common
type
of
mathematical
object,
then,
a
priori,
the
“mysterious
description”
could
be
[among
a
vast
variety
of
possibilities]
one
of
the
following:
(1)
Instead
of
embedding
the
[nonzero
elements
of
the]
base
field
into
H(Π)
via
the
usual
Kummer
map,
one
could
consider
the
embedding
obtained
by
composing
the
usual
Kummer
map
with
the
automorphism
induced
by
some
automorphism
of
the
quotient
Π
G
k
[cf.
the
notation
of
Corollary
1.10,
where
we
take
“Π
X
”
to
be
Π]
which
is
not
of
scheme-theoretic
origin
[cf.,
e.g.,
[NSW],
the
Closing
Remark
preceding
Theorem
12.2.7].
(2)
Alternatively,
instead
of
embedding
the
function
field
of
the
curve
under
consideration
into
H(Π)
via
the
usual
Kummer
map,
one
could
consider
the
embedding
obtained
by
composing
the
usual
Kummer
map
with
the
×
.
automorphism
of
H(Π)
given
by
muliplication
by
some
element
∈
Z
Thus,
in
order
to
ensure
that
such
pathologies
do
not
arise,
it
appears
that
there
is
little
choice
but
to
include
the
ring/scheme-theoretic
models
in
the
common
type
that
one
adopts
as
a
“solution
to
(∗
type
)”,
so
that
one
may
apply
the
“standard
Kummer
map
description”
in
a
simultaneous,
consistent
fashion
to
both
the
coric
data
and
the
log-subject
data.
But
[since
these
models
are
“functorially
obstructed
from
being
subsumed
into
Π”
—
cf.
Remark
3.7.4]
the
inclusion
of
such
ring/scheme-
theoretic
models
amounts
precisely
to
the
“bi-anabelian
approach”
discussed
in
Remark
3.7.3
[cf.,
especially,
Figure
2
of
Remark
3.7.3,
(iv)].
TOPICS
IN
ABSOLUTE
ANABELIAN
GEOMETRY
III
97
(iii)
From
a
“physical”
point
of
view,
it
is
natural
to
think
of
data
that
sat-
isfies
some
sort
of
coricity
—
such
as
the
étale
fundamental
group
Π
—
as
being
×
“massless”,
like
light.
By
comparison,
the
arithmetic
data
“k
”
—
on
which
the
log-Frobenius
functor
acts
non-isomorphically
—
may
be
thought
of
as
being
like
matter
which
has
“weight”.
This
dichotomy
is
reminiscent
of
the
dichotomy
dis-
cussed
in
the
Introduction
to
[Mzk16]
between
“étale-like”
and
“Frobenius-like”
structures.
Thus,
in
summary:
coricity,
“étale-like”
structures
←→
massless,
like
light
“Frobenius-like”
structures
←→
matter
of
positive
mass.
From
this
point
of
view,
the
discussion
of
(i),
(ii)
may
be
summarized
as
follows:
Even
if
the
container
H(Π)
is
massless,
it
one
tries
to
use
it
to
carry
“cargo
of
substantial
weight”,
then
the
resulting
package
[of
container
plus
cargo]
is
no
longer
massless.
On
the
other
hand,
the
very
existence
of
mono-anabelian
algorithms
as
discussed
in
§1,
§2
corresponds,
in
this
analogy,
to
the
“conversion
of
light
into
matter”
[cf.
the
point
of
view
of
Remark
1.9.7]!
(iv)
Relative
to
the
dichotomy
discussed
in
the
Introduction
to
[Mzk16]
between
“étale-like”
and
“Frobenius-like”
structures,
the
problem
observed
in
the
present
paper
with
the
bi-anabelian
approach
may
be
thought
of
as
an
example
of
the
phenomenon
of
the
non-applicability
of
Galois
[i.e.,
“étale-like”]
descent
with
respect
to
“Frobenius-like”
morphisms
[i.e.,
the
existence
of
descent
data
for
a
“Frobenius-
like”
morphism
which
cannot
be
descended
to
an
object
on
the
codomain
of
the
morphism].
In
classical
arithmetic
geometry,
this
phenomenon
may
be
seen,
for
instance,
in
the
non-descendability
of
Galois-invariant
coherent
ideals
with
respect
to
morphisms
such
as
Spec(k[t])
→
Spec(k[t
n
])
[where
n
≥
2
is
an
integer;
k
is
a
field],
or
[cf.
the
discussion
of
“X
×
E
X
”
in
Remark
3.7.2]
the
difference
between
an
integrable
connection
and
an
integrable
connection
equipped
with
a
compatible
Frobenius
action
[e.g.,
of
the
sort
that
arises
from
an
MF
∇
-object].
Remark
3.7.6.
With
regard
to
the
pseudo-mono-anabelian
approach
discussed
in
Remark
3.7.5,
one
may
make
the
following
further
observations.
(i)
In
order
to
carry
out
the
pseudo-mono-anabelian
approach
[or,
a
fortiori,
the
mono-anabelian
approach],
it
is
necessary
to
use
the
full
profinite
étale
fundamental
group
of
a
hyperbolic
orbicurve,
say,
of
strictly
Belyi
type.
That
is
to
say,
if,
for
instance,
one
attempts
to
use
the
geometrically
pro-Σ
fundamental
group
of
a
hyperbolic
curve
[i.e.,
where
Σ
is
a
set
of
primes
which
is
not
equal
to
the
set
of
all
primes],
then
the
crucial
injectivity
of
the
Kummer
map
[cf.
Proposition
1.6,
(i)]
fails
to
hold.
In
particular,
this
failure
of
injectivity
means
that
one
cannot
work
with
the
crucial
additive
structure
on
[the
union
with
{0}
of]
the
image
of
the
Kummer
map.
(ii)
In
a
similar
vein,
if
one
attempts
to
work,
for
instance,
with
the
absolute
Galois
group
of
a
number
field
—
i.e.,
in
the
absence
of
any
geometric
fundamental
group
of
a
hyperbolic
orbicurve
over
the
number
field
—
then,
in
order
to
work
with
Kummer
classes,
one
must
contend
with
the
nontrivial
issue
of
finding
an
appropriate
[profinite]
cyclotome
[i.e.,
copy
of
“
Z(1)”]
to
replace
the
“curve-based
cyclotome
M
X
”
of
Proposition
1.4,
(ii)
[cf.
also
Remark
1.9.5].
98
SHINICHI
MOCHIZUKI
(iii)
Next,
we
observe
that
if
one
attempts
to
construct
“models
of
the
base
field”
via
the
theory
of
“characters
of
qLT-type”
as
in
[Mzk20],
§3
[cf.
also
the
theory
of
[Mzk2],
§4],
then
although
[just
as
was
the
case
with
Kummer
classes]
such
“qLT-models
of
the
base
field”
are
tautologically
Galois-compatible
and
admit
a
coricity
inherited
from
the
coricity
of
Π,
[unlike
the
case
with
Kummer
classes]
the
essential
use
of
p-adic
Hodge
theory
implies
that
the
resulting
“construction
of
the
base
field”
[cf.
the
discussion
of
Remark
1.9.5]
is
incompatible
with
the
operation
of
passing
from
global
[e.g.,
number]
fields
to
local
fields
[i.e.,
does
not
admit
an
analogue
of
the
first
portion
of
Corollary
1.10,
(h)],
hence
also
incompatible
with
the
operation
of
relating
the
resulting
“constructions
of
the
base
field”at
different
localizations
of
a
number
field.
Such
localization
[i.e.,
in
the
terminology
of
§5,
“panalocalization”]
properties
will
play
a
key
role
in
the
theory
of
§5
below.
(iv)
In
the
context
of
(iii),
it
is
interesting
to
note
that
geometrically
pro-Σ
fundamental
groups
as
in
(i)
also
fail
to
be
compatible
with
localization.
Indeed,
even
if
some
sort
of
pro-Σ
analogue
of
the
theory
of
§1
is,
in
the
future,
obtained
for
the
primes
lying
over
prime
numbers
∈
Σ,
such
an
analogue
is
impossible
at
the
primes
lying
over
prime
numbers
∈
Σ
[since,
as
is
easily
verified,
at
such
primes,
the
automorphisms
of
G
k
[notation
of
Corollary
1.10]
that
are
not
of
scheme-theoretic
origin
may
extend,
in
general,
to
automorphisms
of
the
full
arithmetic
[geometri-
cally
pro-Σ]
fundamental
group].
(v)
At
the
time
of
writing,
it
appears
to
be
rather
difficult
to
give
a
mono-
anabelian
“group-theoretic”
algorithm
as
in
Theorem
1.9
in
the
case
of
num-
ber
fields
by
somehow
“gluing
together”
[mono-anabelian,
“group-theoretic”]
al-
gorithms
[cf.
the
approach
via
p-adic
Hodge
theory
discussed
in
(iii)]
applied
at
nonarchimedean
completions
of
the
number
field.
That
is
to
say,
if
one
tries,
for
instance,
to
construct
a
number
field
F
as
a
subset
of
the
product
of
copies
of
F
constructed
at
various
nonarchimedean
completions
of
F
,
then
it
appears
to
be
a
highly
nontrivial
issue
to
reconstruct
the
correspondences
between
the
various
“local
copies”
of
F
.
Indeed,
if
one
attempts
to
work
with
abelianizations
of
local
and
global
Galois
groups
and
apply
class
field
theory
[i.e.,
in
the
fashion
of
[Uchi],
in
the
case
of
function
fields],
then
one
may
only
recover
the
“global
copy”
of
F
×
embedded
in
the
idèles
up
to
an
indeterminacy
that
involves,
in
particular,
various
“solenoids”
[cf.,
e.g.,
[ArTt],
Chapter
Nine,
Theorem
3].
On
the
other
hand,
if
one
attempts
to
work
with
local
and
global
Kummer
classes,
then
one
must
contend
with
the
phenomenon
that
it
is
not
clear
how
to
lift
local
Kummer
classes
to
global
Kummer
classes;
that
is
to
say,
the
indeterminacies
that
occur
for
such
liftings
are
of
a
nature
roughly
reminiscent
of
the
global
Kummer
classes
whose
vanishing
is
implied
by
the
so-called
Leopoldt
Conjecture
[i.e.,
in
its
formulation
concerning
p-adic
localizations
of
units
of
a
number
field],
which
is
unknown
in
general
at
the
time
of
writing.
Remark
3.7.7.
(i)
One
way
to
interpret
the
fact
that
the
log-Frobenius
operation
log
is
not
a
ring
homomorphism
[cf.
the
discussion
of
Remarks
3.7.3,
3.7.4,
3.7.5]
is
to
think
of
TOPICS
IN
ABSOLUTE
ANABELIAN
GEOMETRY
III
99
“log”
as
constituting
a
sort
of
“wall”
that
separates
the
two
“distinct
scheme
theories”
that
occur
before
and
after
its
application.
The
étale
fundamental
groups
that
arise
in
these
“distinct
scheme
theories”
thus
necessarily
correspond
to
dis-
out
tinct,
unrelated
basepoints.
Thus,
if,
for
i
=
1,
2,
G
i
Δ
i
is
a
copy
of
the
out
outer
Galois
action
on
the
geometric
fundamental
group
“G
k
Δ
X
”
of
Theo-
rem
1.9
that
arises
in
one
of
these
two
“distinct
scheme
theories”
separated
by
the
“log-wall”,
then
although
this
log-wall
cannot
be
penetrated
by
ring
structures
[i.e.,
by
“scheme
theory”],
it
can
be
penetrated
by
the
abstract
profinite
group
structure
of
the
G
i
—
cf.
the
Galois-equivariance
of
the
map
“log
k
”
of
Definition
3.1,
(i).
Moreover,
since
the
“abstract
outer
action
pair”
[i.e.,
an
abstract
profinite
group
out
equipped
with
an
outer
action
by
another
abstract
profinite
group]
G
2
Δ
2
is
∼
out
clearly
isomorphic
to
the
composite
abstract
outer
action
pair
G
1
→
G
2
Δ
2
[as
out
well
as,
by
definition,
the
abstract
outer
action
pair
G
1
Δ
1
]
—
i.e.,
out
G
1
Δ
1
∼
→
∼
out
G
1
→
G
2
Δ
2
∼
→
out
G
2
Δ
2
—
we
thus
conclude
that
the
log-wall
can
be
penetrated
by
the
isomorphism
class
out
of
the
abstract
outer
action
pair
G
i
Δ
i
.
G
1
∼
→
log
Δ
1
G
2
out
out
∼
→
log
log
Δ
2
(ii)
Once
one
has
made
the
observations
made
in
(i),
it
is
natural
to
proceed
to
consider
what
sort
of
“additional
data”
may
be
shared
on
both
sides
of
the
log-wall.
out
Typically,
“purely
group-theoretic
structures”
constructed
from
“G
i
Δ
i
”
serve
as
natural
containers
for
such
additional
data
[cf.,
e.g.,
the
discussion
of
Remark
3.7.5].
Thus,
the
additional
data
may
be
thought
as
some
sort
of
a
choice
[cf.
the
dotted
arrows
in
the
diagram
below]
among
various
possibilities
[cf.
the
“
’s”
in
the
diagram
below]
housed
in
such
a
group-theoretic
container.
log
log
log
log
From
this
point
of
view:
100
SHINICHI
MOCHIZUKI
The
fundamental
difference
that
distinguishes
the
pseudo-mono-anabelian
approach
discussed
in
Remark
3.7.5
from
the
mono-anabelian
approach
is
the
issue
of
whether
this
“choice”
is
specified
in
terms
that
depend
on
the
scheme
theory
that
gives
rise
to
the
choice
[i.e.,
the
pseudo-mono-
anabelian
case]
or
not
[i.e.,
the
mono-anabelian
case,
in
which
the
choice
may
be
specified
in
language
that
depends
only
on
the
abstract
group
out
structure
of,
say,
“G
i
Δ
i
”].
In
fact,
the
discussion
in
Remark
3.7.5,
(ii)
[cf.
also
Figs.
1,
2
of
Remark
3.7.3,
(iv)],
may
be
depicted
via
a
similar
illustration
to
the
above
illustration
of
the
“log-
wall”
in
which
the
“log-wall”
is
replaced
by
a
“model-wall”
separating
“reference
models”
from
“log-subject
copies”
of
such
models.
In
Remark
3.7.5,
(ii),
special
attention
was
given
to
the
situation
in
which
the
“additional
data”
consists
of
the
“additive
structure”
on
the
image
of
the
Kummer
map.
When
the
Δ
i
’s
of
(i)
are
given
by
the
birational
geometric
fundamental
groups
“Δ
η
X
”
of
Theorem
1.11,
another
example
of
such
“additional
data”
in
which
the
specification
of
the
“choice”
depends
on
“scheme
theory”
[and
hence
cannot,
at
least
a
priori,
be
shared
on
both
sides
of
the
log-wall]
is
given
by
the
specification
of
some
finite
collection
of
closed
points
corresponding
to
the
cusps
of
some
affine
hyperbolic
curve
that
lies
in
some
given
scheme
theory
[cf.
Remark
1.11.5].
(iii)
With
regard
to
the
issue
of
“specifying
some
finite
collection
of
closed
points
corresponding
to
the
cusps
of
an
affine
hyperbolic
curve”
discussed
in
the
final
portion
of
(ii),
we
note
that
in
certain
special
cases,
a
“purely
group-theoretic”
specification
is
in
fact
possible.
For
instance,
if,
in
the
notation
of
Theorem
1.11,
X
is
a
hyperelliptic
curve
whose
unique
nontrivial
k-automorphism
is
given
by
its
hyperelliptic
involution,
then
the
set
of
points
fixed
by
the
hyperelliptic
involution
constitutes
such
an
example
in
which
a
“purely
group-theoretic”
specification
can
be
made
by
considering
the
conjugacy
classes
of
inertia
groups
“I
x
”
fixed
by
the
unique
nontrivial
outer
automorphism
of
Δ
η
X
that
commutes
with
the
given
outer
action
of
G
k
on
Δ
η
X
.
(iv)
The
“log-wall”
discussed
in
(i)
is
reminiscent
of
the
constant
indeterminacy
arising
from
morphisms
of
Frobenius
type
[i.e.,
which
thus
constitute
a
“wall”
that
cannot
be
penetrated
by
constant
rigidity]
in
the
theory
of
the
étale
theta
function
[cf.
[Mzk18],
Corollary
5.12
and
the
following
remarks],
as
well
as
of
the
subtleties
that
arise
from
the
Frobenius
morphism
in
the
context
of
anabelian
geometry
in
positive
characteristic
[cf.,
e.g.,
[Stix]].
Remark
3.7.8.
Many
of
the
arguments
in
the
various
remarks
following
Corollaries
3.6,
3.7
are
not
formulated
entirely
rigorously.
Thus,
in
the
future,
it
is
quite
possible
that
certain
of
the
obstacles
pointed
out
in
these
remarks
can
be
overcome.
Nevertheless,
we
presented
these
remarks
in
the
hope
that
they
could
aid
in
elucidating
the
content
of
and
motivation
[from
the
point
of
view
of
the
author]
behind
the
various
rigorously
formulated
results
of
the
present
paper.
TOPICS
IN
ABSOLUTE
ANABELIAN
GEOMETRY
III
101
Section
4:
Archimedean
Log-Frobenius
Compatibility
In
the
present
§4,
we
present
an
archimedean
version
[cf.
Corollary
4.5]
of
the
theory
of
§3,
i.e.,
we
interpret
the
theory
of
§2
in
terms
of
a
certain
compatibility
with
the
“log-Frobenius
functor”.
Definition
4.1.
(i)
Let
k
be
a
CAF
[cf.
§0].
Write
O
k
⊆
k
for
the
subset
of
elements
of
absolute
value
≤
1,
O
k
×
⊆
O
k
for
the
subgroup
of
units
[i.e.,
elements
of
absolute
value
equal
to
1
—
cf.
[Mzk17],
Definition
3.1,
(ii)],
O
k
⊆
O
k
for
the
multiplicative
monoid
of
def
nonzero
elements,
and
k
∼
k
×
=
k\{0}
for
the
universal
covering
of
k
×
.
Thus,
k
∼
k
×
is
uniquely
determined,
up
to
unique
isomorphism,
as
a
pointed
topological
space
and,
moreover,
[as
is
well-known]
may
be
constructed
explicitly
by
considering
homotopy
classes
of
paths
on
k
×
;
moreover,
the
pointed
topological
space
k
∼
admits
a
natural
topological
group
structure,
determined
by
the
topological
group
structure
of
k
×
.
Note
that
the
“inverse”
of
the
exponential
map
k
→
k
×
[given
by
the
usual
power
series]
determines
an
isomorphism
of
topological
groups
∼
log
k
:
k
∼
→
k
—
which
we
shall
refer
to
as
the
logarithm
associated
to
k.
Next,
let
X
ell
be
an
elliptically
admissible
Aut-holomorphic
orbispace
[cf.
Definition
2.1,
(i);
Re-
mark
2.1.1].
We
shall
refer
to
as
a
[k-]Kummer
structure
on
X
ell
any
isomorphism
of
topological
fields
∼
def
κ
k
:
k
→
A
X
ell
=
A
X
ell
{0}
—
where
we
write
A
X
ell
for
the
“A
p
”
of
Corollary
2.7,
(e)
[equipped
with
various
topological
and
algebraic
structures],
which
may
be
identified
[hence
considered
as
∼
an
object
that
is
independent
of
“p”]
via
the
various
isomorphisms
“A
p
→
A
p
”
of
Corollary
2.7,
(e),
together
with
the
functoriality
of
the
algorithms
of
Corollary
2.7.
Note
that
k,
k
×
,
k
∼
,
and
A
X
ell
are
equipped
with
natural
Aut-holomorphic
structures,
with
respect
to
which
κ
k
determines
co-holomorphicizations
between
k
and
A
X
ell
,
as
well
as
between
k
∼
and
A
X
ell
;
moreover,
these
co-holomorphicizations
are
compatible
with
log
k
.
Next,
let
T
∈
{TF,
TCG,
TLG,
TM,
TH,
TH}
—
where
TF,
TCG,
TLG,
TM
as
in
Definition
3.1,
(i),
and
we
write
TH
for
the
category
of
connected
Aut-holomorphic
orbispaces
and
morphisms
of
Aut-
holomorphic
orbispaces
[cf.
Remark
2.1.1],
and
TH
102
SHINICHI
MOCHIZUKI
for
the
category
of
connected
Aut-holomorphic
groups
[i.e.,
Aut-holomorphic
spaces
equipped
with
a
topological
group
structure
such
that
both
the
Aut-holomorphic
and
topological
group
structures
arise
from
a
single
connected
complex
Lie
group
structure]
and
homomorphisms
of
Aut-holomorphic
groups.
If
T
is
equal
to
TF
(respectively,
TCG;
TLG;
TM;
TH;
TH),
then
let
M
k
∈
Ob(T)
be
the
object
determined
by
k
(respectively,
the
object
determined
by
O
k
×
;
the
object
deter-
mined
by
k
×
;
the
object
determined
by
O
k
;
any
object
of
TH
equipped
with
a
co-holomorphicization
κ
M
k
:
M
k
→
A
X
ell
;
any
object
of
TH
equipped
with
an
Aut-holomorphic
homomorphism
κ
M
k
:
M
k
→
A
X
ell
(⊆
A
X
ell
)
[relative
to
the
mul-
tiplicative
structure
of
A
X
ell
]);
if
T
=
TH,
TH
and
κ
k
is
a
k-Kummer
structure
on
X
ell
,
then
write
κ
M
k
:
M
k
→
A
X
ell
for
the
restriction
of
κ
k
to
M
k
⊆
k.
We
shall
refer
to
as
a
model
Aut-holomorphic
T-pair
any
collection
of
data
(a),
(b),
(c)
of
the
following
form:
(a)
the
elliptically
admissible
Aut-holomorphic
orbispace
X
ell
,
(b)
the
object
M
k
∈
Ob(T),
(c)
the
datum
κ
M
k
:
M
k
→
A
X
ell
.
Also,
we
shall
refer
to
the
datum
κ
M
k
of
(c)
as
the
Kummer
structure
of
the
model
κ
Aut-holomorphic
T-pair;
we
shall
often
use
the
abbreviated
notation
(X
ell
M
k
)
for
this
collection
of
data
(a),
(b),
(c).
κ
(ii)
We
shall
refer
to
any
collection
of
data
(X
M
)
consisting
of
an
elliptically
admissible
Aut-holomorphic
orbispace
X,
an
object
M
∈
Ob(T),
and
a
datum
κ
κ
M
:
M
→
A
X
,
which
we
shall
refer
to
as
the
Kummer
structure
of
(X
M
),
as
an
Aut-holomorphic
T-pair
if
the
following
conditions
are
satisfied:
(a)
κ
M
is
a
continuous
map
between
the
underlying
topological
spaces
whenever
T
=
TH;
(b)
κ
M
is
a
collection
of
continuous
maps
from
open
subsets
of
the
underlying
topological
space
of
M
to
the
underlying
topological
space
of
A
X
whenever
T
=
TH;
κ
(c)
for
some
model
Aut-holomorphic
T-pair
(X
ell
M
k
)
[where
the
notation
is
as
∼
in
(i)],
there
exist
an
isomorphism
X
ell
→
X
of
objects
of
TH
and
an
isomorphism
∼
M
k
→
M
of
objects
of
T
that
are
compatible
with
the
respective
Kummer
structures
κ
M
k
:
M
k
→
A
X
ell
,
κ
M
:
M
→
A
X
.
In
this
situation,
we
shall
refer
to
X
as
the
structure-orbispace
and
to
M
as
the
arithmetic
data
of
the
Aut-holomorphic
T-pair
κ
(X
M
);
if,
in
this
situation,
the
structure-orbispace
X
arises
from
a
hyperbolic
orbicurve
which
is
of
strictly
Belyi
type
[cf.
Remark
2.8.3],
then
we
shall
refer
to
κ
the
Aut-holomorphic
T-pair
(X
M
)
as
being
of
strictly
Belyi
type.
A
morphism
of
Aut-holomorphic
T-pairs
κ
κ
φ
:
(X
1
M
1
)
→
(X
2
M
2
)
consists
of
a
morphism
of
objects
φ
M
:
M
1
→
M
2
of
T,
together
with
a
compatible
[relative
to
the
respective
Kummer
structures]
finite
étale
morphism
φ
X
:
X
1
→
X
2
of
TH;
if,
in
this
situation,
φ
M
(respectively,
φ
X
)
is
an
isomorphism,
then
we
shall
refer
to
φ
as
a
T-isomorphism
(respectively,
structure-isomorphism).
(iii)
Write
C
T
hol
TOPICS
IN
ABSOLUTE
ANABELIAN
GEOMETRY
III
103
for
the
category
whose
objects
are
the
Aut-holomorphic
T-pairs
and
whose
mor-
phisms
are
the
morphisms
of
Aut-holomorphic
T-pairs.
Also,
we
shall
use
the
same
notation,
except
with
“C”
replaced
by
C
(respectively,
C;
C)
to
denote
the
various
subcategories
determined
by
the
T-isomorphisms
(respec-
tively,
structure-isomorphisms;
isomorphisms);
we
shall
use
the
same
notation,
with
“hol”
replaced
by
hol-sB
to
denote
the
various
full
subcategories
determined
by
the
objects
of
strictly
Belyi
type.
Since
[in
the
notation
of
(i)]
the
formation
of
O
k
(respectively,
k
×
;
O
k
×
;
O
k
×
)
from
k
(respectively,
O
k
;
O
k
;
k
×
)
is
clearly
intrinsically
defined
[i.e.,
depends
only
on
the
“input
data
of
an
object
of
T”],
we
thus
obtain
natural
functors
hol
hol
→
C
TM
;
C
TF
hol
hol
C
TM
→
C
TLG
;
hol
hol
C
TM
→
C
TCG
;
hol
hol
C
TLG
→
C
TCG
—
i.e.,
by
taking
the
multiplicative
monoid
of
nonzero
elements
of
absolute
value
≤
1
of
the
arithmetic
data
[i.e.,
nonzero
elements
of
the
closure
of
the
set
of
elements
a
such
that
a
n
→
0
as
n
→
∞],
the
associated
groupification
M
gp
of
the
arithmetic
data
M
,
the
subgroup
of
invertible
elements
M
×
of
the
arithmetic
data
M
,
or
the
maximal
compact
subgroup
of
the
arithmetic
data.
Finally,
we
shall
write
TH
⊇
EA
⊇
EA
sB
for
the
subcategories
determined,
respectively,
by
the
elliptically
admissible
hyper-
bolic
orbicurves
over
CAF’s
and
the
finite
étale
morphisms,
and
by
the
elliptically
admissible
hyperbolic
orbicurves
of
strictly
Belyi
type
over
CAF’s
and
the
finite
étale
morphisms;
also,
we
shall
use
the
same
notation,
except
with
“EA”
replaced
by
EA
to
denote
the
subcategory
determined
by
the
isomorphisms.
Thus,
for
κ
T
∈
{TF,
TCG,
TLG,
TM,
TH,
TH},
the
assignment
(X
M
)
→
X
determines
various
compatible
natural
functors
C
T
hol
→
EA
[as
well
as
double
underlined
versions
of
these
functors].
(iv)
Observe
that
[in
the
notation
of
(i)]
the
field
structure
of
k
determines,
via
the
inverse
morphism
to
log
k
,
a
structure
of
topological
field
on
the
topological
group
k
∼
;
moreover,
κ
k
determines
a
k
∼
-Kummer
structure
on
X
ell
∼
κ
k
∼
:
k
∼
→
A
X
ell
which
may
be
uniquely
characterized
[i.e.,
among
the
two
k
∼
-Kummer
structures
on
X
ell
]
by
the
property
that
the
co-holomorphicization
determined
by
κ
k
∼
coincides
with
the
co-holomorphicization
determined
by
composing
the
composite
of
natural
maps
k
∼
k
×
→
k
with
the
co-holomorphicization
determined
by
κ
k
.
In
par-
ticular,
[cf.
(i)]
the
co-holomorphicizations
determined
by
κ
k
,
κ
k
∼
are
compatible
104
SHINICHI
MOCHIZUKI
with
log
k
.
Since
the
various
operations
applied
here
to
construct
this
field
struc-
ture
on
k
∼
[such
as,
for
instance,
the
power
series
used
to
define
log
k
]
are
clearly
intrinsically
defined
[cf.
the
natural
functors
defined
in
(iii)],
we
thus
obtain
that
the
construction
that
assigns
(the
topological
field
k,
with
its
Kummer
structure
κ
k
)
→
(the
topological
field
k
∼
,
with
its
Kummer
structure
κ
k
∼
)
determines
a
natural
functor
hol
hol
→
C
TF
log
TF,TF
:
C
TF
—
which
we
shall
refer
to
as
the
log-Frobenius
functor.
Since
log
k
determines
a
functorial
isomorphism
between
the
fields
k,
k
∼
,
it
follows
immediately
that
the
functor
log
TF,TF
is
isomorphic
to
the
identity
functor
[hence,
in
particular,
is
an
equivalence
of
categories].
By
composing
log
TF,TF
with
the
various
natural
functors
defined
in
(iii),
we
also
obtain,
for
T
∈
{TLG,
TCG,
TM},
a
functor
hol
log
TF,T
:
C
TF
→
C
T
hol
—
which
[by
abuse
of
terminology]
we
shall
also
refer
to
as
“the
log-Frobenius
functor”.
In
a
similar
vein,
the
assignments
(the
topological
field
k,
with
its
Kummer
structure
κ
k
)
→
(the
Aut-holomorphic
space
k
×
,
with
its
Kummer
structure
[κ
k
|
k
×
])
(the
topological
field
k,
with
its
Kummer
structure
κ
k
)
→
(the
Aut-holomorphic
space
k
∼
,
with
its
Kummer
structure
[κ
k
∼
])
—
where
the
[−]’s
denote
the
associated
co-holomorphicizations;
the
phrase
“the
Aut-holomorphic
space”
should,
strictly
speaking,
be
interpreted
as
meaning
“the
Aut-holomorphic
space
determined
by”
—
determine
natural
functors
hol
hol
→
C
TH
;
λ
×
:
C
TF
hol
hol
λ
∼
:
C
TF
→
C
TH
together
with
diagrams
of
functors
hol
C
TF
⏐
⏐
∼
λ
hol
C
TH
log
TF,TF
−→
hol
C
TF
⏐
⏐
×
λ
hol
C
TF
⏐
ι
⏐
⏐
×
⏐
∼
λ
×
λ
=
hol
C
TH
hol
C
TH
ι
log
—
where
we
write
ι
log
:
λ
×
◦
log
TF,TF
→
λ
∼
for
the
natural
transformation
in-
duced
by
the
natural
inclusion
“(k
∼
)
×
→
k
∼
”
and
ι
×
:
λ
∼
→
λ
×
for
the
natural
transformation
induced
by
the
natural
map
“k
∼
k
×
”.
Finally,
we
note
that
the
fields
“k
∼
”
obtained
by
the
above
construction
[i.e.,
the
arithmetic
data
of
the
objects
in
the
image
of
the
log-Frobenius
functor
log
TF,TF
]
are
equipped
with
a
nat-
ural
“subquotient
compactum”
—
i.e.,
the
compact
subset
“O
k
×
⊆
k
×
”
that
lies
in
TOPICS
IN
ABSOLUTE
ANABELIAN
GEOMETRY
III
105
the
natural
quotient
“k
∼
k
×
”
—
which
we
shall
refer
to
as
the
pre-log-shell
of
κ
log
arith
TF,TF
((X
M
))
κ
λ
(X
M
)
κ
hol
κ
—
where
(X
M
)
∈
Ob(C
TF
);
λ
(X
is
a
subquotient
of
the
arithmetic
data
M
)
κ
log
arith
TF,TF
((X
M
))
of
the
object
determined
by
applying
the
log-Frobenius
functor
κ
log
TF,TF
to
the
object
(X
M
).
(v)
Write
LinHol
[i.e.,
“linear
holomorphic”]
for
the
category
whose
objects
are
pairs
κ
X,
X
A
X
consisting
of
an
object
X
∈
Ob(EA),
together
with
the
tautological
Kummer
map
∼
A
X
→
A
X
[i.e.,
given
by
the
identity
on
the
object
of
TF
determined
by
A
X
]
—
all
of
which
is
to
be
understood
as
constructed
via
the
algorithms
of
Corollary
2.7
[cf.
Remark
3.1.2]
—
and
whose
morphisms
are
the
morphisms
induced
by
the
[finite
étale]
morphisms
of
EA
[cf.
the
functorial
algorithms
of
Corollary
2.7].
Thus,
we
obtain
a
natural
functor
κ
LH
EA
−→
LinHol
which
[as
is
easily
verified]
is
an
equivalence
of
categories,
a
quasi-inverse
for
which
is
given
by
the
natural
projection
functor
LinHol
→
EA.
Remark
4.1.1.
The
topological
monoid
“O
k
”
associated
to
a
CAF
k
[cf.
Defini-
tion
4.1,
(i)]
is
essentially
the
data
used
to
construct
the
archimedean
Frobenioids
of
[Mzk17],
Example
3.3,
(ii).
Remark
4.1.2.
Although,
to
simplify
the
discussion,
we
have
chosen
to
require
that
the
structure-orbispace
always
be
elliptically
admissible,
and
that
the
base
field
always
be
a
CAF,
many
aspects
of
the
theory
of
the
present
§4
may
be
general-
ized
to
accommodate
“structure-orbispaces”
that
are
Aut-holomorphic
orbispaces
that
arise
from
more
general
hyperbolic
orbicurves
[cf.,
e.g.,
Propositions
2.5,
2.6;
Remark
2.6.1]
over
arbitrary
archimedean
fields
[i.e.,
either
CAF’s
or
RAF’s
—
cf.
§0].
Such
generalizations,
however,
are
beyond
the
scope
of
the
present
paper.
Proposition
4.2.
(First
Properties
of
Aut-Holomorphic
Pairs)
κ
κ
(i)
Let
T
∈
{TM,
TF,
TLG,
TCG};
(X
M
),
(X
∗
M
∗
)
∈
Ob(C
T
hol
).
Then
the
natural
functor
of
Definition
4.1,
(iii),
induces
a
bijection
[cf.
Proposition
3.2,
(iv)]
κ
κ
∼
Isom
C
hol
((X
M
),
(X
∗
M
∗
))
→
Isom
EA
(X,
X
∗
)
T
hol-sB
=
on
sets
of
isomorphisms.
In
particular,
the
categories
EA,
C
T
hol
=
C
hol
T
,
C
T
hol-sB
are
id-rigid.
C
T
∼
(ii)
The
equivalence
of
categories
κ
LH
:
EA
→
LinHol
of
Definition
4.1,
(v)
—
i.e.,
the
functorial
algorithms
of
Corollary
2.7
—
determines
a
natural
[1-
]factorization
[cf.
Proposition
3.2,
(v)]
hol
C
TF
−→
hol
C
TM
log
TM,T
−→
C
T
hol
106
SHINICHI
MOCHIZUKI
—
where
T
∈
{TF,
TLG,
TCG,
TM};
the
first
arrow
is
the
natural
functor
of
Defini-
hol
→
C
T
hol
of
Definition
tion
4.1,
(iii)
—
of
the
log-Frobenius
functors
log
TF,T
:
C
TF
4.1,
(iv).
Moreover,
[when
T
∈
{TF,
TM}]
the
functor
log
T,T
is
isomorphic
to
the
identity
functor
[hence,
in
particular,
is
an
equivalence
of
categories].
Proof.
The
bijectivity
portion
of
assertion
(i)
follows
immediately
from
the
required
compatibility
of
morphisms
of
C
T
hol
with
the
Kummer
structures
of
the
objects
in-
volved
[cf.
also
the
functorial
algorithms
of
Corollary
2.7].
To
verify
the
id-rigidity
of
EA,
it
suffices
to
observe
that
for
any
object
X
∈
Ob(EA),
which
necessarily
arises
from
some
elliptically
admissible
hyperbolic
orbicurve
X
over
a
CAF,
the
full
subcategory
of
EA
consisting
of
objects
that
map
to
X
may,
by
Corollary
2.3,
(i)
[cf.
also
[Mzk14],
Lemma
1.3,
(iii)],
be
identified
with
the
category
of
finite
étale
R-localizations
“Loc
R
(X)”
[i.e.,
the
subcategory
of
the
category
of
finite
étale
localizations
“Loc(X)”
of
[Mzk10],
§2,
obtained
by
considering
the
R-linear
mor-
phisms].
Thus,
the
id-rigidity
of
EA
follows
immediately
from
the
slimness
assertion
of
Lemma
4.3
below.
In
light
of
the
bijectivity
portion
of
assertion
(i),
the
id-rigidity
hol-sB
=
C
hol-sB
follows
in
a
similar
fashion.
This
of
the
categories
C
T
hol
=
C
hol
T
,
C
T
T
completes
the
proof
of
assertion
(i).
Assertion
(ii)
follows
immediately
from
the
definitions
[and
the
functorial
algorithms
of
Corollary
2.7].
Remark
4.2.1.
Note
that,
unlike
the
case
with
Proposition
3.2,
(iv),
the
id-
rigidity
portion
of
Proposition
4.2,
(i),
is
[as
is
easily
verifed]
false
for
the
“C”
and
“C”
versions
of
C
T
hol
,
C
T
hol-sB
.
The
following
result
is
well-known.
Lemma
4.3.
(Slimness
of
Archimedean
Fundamental
Groups)
Let
X
be
a
hyperbolic
orbicurve
over
an
archimedean
field
k
X
.
Then
the
étale
fundamental
group
Π
X
of
X
is
slim.
Proof.
Let
k
X
be
an
algebraic
closure
of
k
X
.
Thus,
we
have
an
exact
sequence
of
profinite
groups
1
→
Δ
X
→
Π
X
→
G
→
1
def
def
[where
Δ
X
=
π
1
(X
×
k
X
k
X
);
G
=
Gal(k
X
/k
X
)].
Since
Δ
X
is
slim
[cf.,
e.g.,
[Mzk20],
Proposition
2.3,
(i)],
it
suffices
to
consider
the
case
where
there
exists
an
element
σ
∈
Π
X
that
maps
to
a
nontrivial
element
σ
G
∈
G
∼
=
Z/2Z
and,
moreover,
commutes
with
some
open
subgroup
H
⊆
Π
X
.
We
may
assume
without
loss
of
generality
that
H
⊆
Δ
X
,
and,
moreover,
that
H
corresponds
to
a
finite
étale
covering
of
X
×
k
X
k
X
which
is
a
hyperbolic
curve
of
genus
≥
2.
In
particular,
by
replacing
X
by
the
finite
étale
covering
of
X
determined
by
the
open
subgroup
of
Π
X
generated
by
H
and
σ,
we
may
assume
that
σ
lies
in
the
center
of
Π
X
,
and,
moreover,
that
X
is
a
hyperbolic
curve
of
genus
≥
2.
In
particular,
by
filling
in
the
cusps
of
X,
we
may
assume
further
that
X
is
proper.
Now
if
l
is
any
prime
number,
then
the
first
Chern
class
of,
say,
the
canonical
bundle
of
X
determines
a
generator
of
H
2
(X
×
k
X
k
X
,
Q
l
(1))
∼
=
H
2
(Δ
X
,
Q
l
(1))
[where
the
“(1)”
denotes
a
Tate
∼
twist],
hence
an
isomorphism
of
G-modules
H
2
(Δ
X
,
Q
l
)
→
Q
l
(−1).
In
particular,
TOPICS
IN
ABSOLUTE
ANABELIAN
GEOMETRY
III
107
it
follows
that
σ
acts
nontrivially
on
H
2
(Δ
X
,
Q
l
),
in
contradiction
to
the
fact
that
σ
lies
in
the
center
of
Π
X
.
This
contradiction
completes
the
proof
of
Lemma
4.3.
Lemma
4.4.
(Topological
Distinguishability
of
Additive
and
Multi-
plicative
Structures)
Let
k
be
a
CAF
[cf.
Definition
4.1,
(i)].
Then
[in
the
notation
of
Definition
4.1,
(i)]
no
composite
of
the
form
k
×
α
−→
(k
∼
)
×
→
k
∼
k
×
—
where
the
“×”
of
“(k
∼
)
×
”
is
relative
to
the
field
structure
of
k
∼
[cf.
Definition
4.1,
(iv)];
α
is
an
isomorphism
of
topological
groups;
“→”
is
the
natural
inclusion;
“”
is
the
natural
map
—
is
bijective.
Proof.
Indeed,
the
non-injectivity
of
k
∼
k
×
implies
that
the
composite
under
consideration
fails
to
be
injective.
Corollary
4.5.
(Aut-Holomorphic
Mono-anabelian
Log-Frobenius
Com-
patibility)
Write
def
def
hol
X
=
C
hol
T
=
C
T
;
E
=
EA;
def
hol
N
=
C
TH
—
where
[in
the
notation
of
Definition
3.1]
T
∈
{TM,
TF}.
Consider
the
diagram
of
categories
D
...
...
X
log
−→
X
⏐
⏐
id
id
+1
log
−→
X
...
id
−1
...
X
⏐
⏐
⏐
⏐
∼
λ
×
λ
N
⏐
⏐
E
⏐
⏐
κ
LH
LinHol
⏐
⏐
E
—
where
we
use
the
notation
“log”,
“λ
×
”,
“λ
∼
”
for
the
arrows
“log
T,T
”,
“λ
×
”,
“λ
∼
”
of
Definition
4.1,
(iv)
[cf.
also
Proposition
4.2,
(ii)];
we
employ
the
conven-
tions
of
Corollary
3.6
concerning
subdiagrams
of
D;
we
write
L
for
the
countably
ordered
set
determined
by
[cf.
§0]
the
infinite
linear
oriented
graph
Γ
opp
D
≤1
[so
the
elements
of
L
correspond
to
vertices
of
the
first
row
of
D]
and
L
†
=
L
∪
{}
def
108
SHINICHI
MOCHIZUKI
for
the
ordered
set
obtained
by
appending
to
L
a
formal
symbol
[which
we
think
of
as
corresponding
to
the
unique
vertex
of
the
second
row
of
D]
such
that
<
,
for
all
∈
L;
id
denotes
the
identity
functor
at
the
vertex
∈
L.
Then:
(i)
For
n
=
4,
5,
6,
D
≤n
admits
a
natural
structure
of
core
on
D
≤n−1
.
That
is
to
say,
loosely
speaking,
E,
LinHol
“form
cores”
of
the
functors
in
D.
(ii)
The
assignments
κ
κ
X,
X
A
X
→
(X
A
X
),
κ
(X
A
X
)
[where
we
write
“A
”
for
the
monoid
of
nonzero
elements
of
absolute
value
≤
1
of
the
CAF
given
by
“A”]
determine
[i.e.,
for
each
choice
of
T]
a
natural
“forgetful”
functor
φ
LH
LinHol
−→
X
which
is
an
equivalence
of
categories,
a
quasi-inverse
for
which
is
given
by
the
composite
π
LH
:
X
→
LinHol
of
the
natural
projection
functor
X
→
E
with
∼
κ
LH
:
E
→
LinHol;
write
η
LH
:
φ
LH
◦
π
LH
→
id
X
for
the
tautological
isomorphism
arising
from
the
definitions
[cf.
Definition
4.1,
(i),
(ii)].
Moreover,
φ
LH
gives
rise
to
a
telecore
structure
T
LH
on
D
≤4
,
whose
underlying
diagram
of
categories
we
denote
by
D
LH
,
by
appending
to
D
≤5
telecore
edges
...
φ
+1
...
X
LinHol
⏐
⏐
φ
log
−→
LinHol
φ
−1
...
X
...
log
X
−→
φ
−→
X
from
the
core
LinHol
to
the
various
copies
of
X
in
D
≤2
given
by
copies
of
φ
LH
,
which
we
denote
by
φ
,
for
∈
L
†
.
That
is
to
say,
loosely
speaking,
φ
LH
determines
0
]
for
the
path
a
telecore
structure
on
D
≤4
.
Finally,
for
each
∈
L
†
,
let
us
write
[β
1
on
Γ
D
LH
of
length
0
at
and
[β
]
for
the
path
on
Γ
D
LH
of
length
∈
{4,
5}
[i.e.,
depending
on
whether
or
not
=
]
that
starts
from
,
descends
[say,
via
λ
×
]
to
the
core
vertex
“LinHol”,
and
returns
to
via
the
telecore
edge
φ
.
Then
the
collection
of
natural
transformations
−1
−1
{η
,
η
,
η
,
η
}
∈L,∈L
†
—
where
we
write
η
for
the
identity
natural
transformation
from
the
arrow
φ
:
LinHol
→
X
to
the
composite
arrow
id
◦
φ
:
LinHol
→
X
and
∼
η
:
(D
LH
)
[β
1
]
→
(D
LH
)
[β
0
]
for
the
isomorphism
arising
from
η
LH
—
generate
a
contact
structure
H
LH
on
the
telecore
T
LH
.
(iii)
The
natural
transformations
ι
log,
:
λ
×
◦
id
◦
log
→
λ
∼
◦
id
+1
,
ι
×
:
λ
∼
→
λ
×
TOPICS
IN
ABSOLUTE
ANABELIAN
GEOMETRY
III
109
[cf.
Definition
4.1,
(iv)]
belong
to
a
family
of
homotopies
on
D
≤3
that
determines
on
D
≤3
a
structure
of
observable
S
log
on
D
≤2
and,
moreover,
is
compatible
with
the
families
of
homotopies
that
constitute
the
core
and
telecore
structures
of
(i),
(ii).
(iv)
The
diagram
of
categories
D
≤2
does
not
admit
a
structure
of
core
on
D
≤1
which
[i.e.,
whose
constituent
family
of
homotopies]
is
compatible
with
[the
constituent
family
of
homotopies
of
]
the
observable
S
log
of
(iii).
Moreover,
the
telecore
structure
T
LH
of
(ii),
the
contact
structure
H
LH
of
(ii),
and
the
observable
S
log
of
(iii)
are
not
simultaneously
compatible
[but
cf.
Remark
3.7.3,
(ii)].
(v)
The
unique
vertex
of
the
second
row
of
D
is
a
nexus
of
Γ
D
.
More-
over,
D
is
totally
-rigid,
and
the
natural
action
of
Z
on
the
infinite
linear
oriented
graph
Γ
D
≤1
extends
to
an
action
of
Z
on
D
by
nexus-classes
of
self-
equivalences
of
D.
Finally,
the
self-equivalences
in
these
nexus-classes
are
com-
patible
with
the
families
of
homotopies
that
constitute
the
cores
and
observ-
able
of
(i),
(iii);
these
self-equivalences
also
extend
naturally
[cf.
the
technique
of
extension
applied
in
Definition
3.5,
(vi)]
to
the
diagram
of
categories
[cf.
Definition
3.5,
(iv),
(a)]
that
constitutes
the
telecore
of
(ii),
in
a
fashion
that
is
compatible
with
both
the
family
of
homotopies
that
constitutes
this
telecore
structure
[cf.
Definition
3.5,
(iv),
(b)]
and
the
contact
structure
H
LH
of
(ii).
Proof.
Assertions
(i),
(ii)
are
immediate
from
the
definitions
[and
the
functorial
algorithms
of
Corollary
2.7]
—
cf.
also
the
proofs
of
Corollary
3.6,
(i),
(ii).
Next,
×
we
consider
assertion
(iii).
If,
for
∈
L,
one
denotes
by
“k
”
the
arithmetic
data
of
type
TLG
[which
we
may
be
obtained
from
an
arithmetic
data
of
type
T
∈
{TM,
TF}
via
the
natural
functors
of
Definition
4.1,
(iii)]
of
a
“typical
object”
of
the
copy
of
X
at
the
vertex
of
D
≤1
,
then
ι
×
“applied
at
the
vertex
”
corresponds
×
∼
to
the
natural
surjection
k
k
,
while
ι
log,
corresponds
to
the
natural
inclusion
×
×
×
∼
k
→
k
+1
,
where
we
think
of
k
as
being
obtained
from
k
+1
via
the
application
of
log.
In
particular,
by
letting
∈
L
vary
and
composing
these
natural
surjections
and
inclusions,
we
obtain
a
diagram
...
×
×
∼
×
∼
∼
k
→
k
+1
k
+1
→
k
+2
k
+2
→
→
k
...
[which
is
compatible
with
the
various
Kummer
structures
—
cf.
Remark
4.5.1,
(i),
hol
in
Definition
4.1,
(i),
(ii),
(iii)].
The
paths
on
[the
below;
the
definition
of
C
TH
oriented
graph
corresponding
to]
this
diagram
may
be
classified
into
four
types,
which
correspond
[by
composing,
in
an
alternating
fashion,
various
pull-backs
of
“ι
log,
”
with
various
pull-backs
of
ι
×
]
to
homotopies
on
D
≤3
,
as
follows
[cf.
the
notational
conventions
of
the
proof
of
Corollary
3.6]:
×
∼
(1)
the
path
corresponding
to
the
composite
“k
→
k
+n
”,
which
yields
a
×
n
∼
homotopy
for
pairs
of
paths
([λ
]
◦
[id
]
◦
[log]
◦
[γ],
[λ
]
◦
[id
+n
]
◦
[γ])
×
×
(2)
the
path
corresponding
to
the
composite
“k
→
k
+n
”,
which
yields
a
×
n
×
homotopy
for
pairs
of
paths
([λ
]
◦
[id
]
◦
[log]
◦
[γ],
[λ
]
◦
[id
+n
]
◦
[γ])
∼
∼
→
k
+n
”,
which
yields
a
(3)
the
path
corresponding
to
the
composite
“k
∼
n
∼
homotopy
for
pairs
of
paths
([λ
]
◦
[id
]
◦
[log]
◦
[γ],
[λ
]
◦
[id
+n
]
◦
[γ])
110
SHINICHI
MOCHIZUKI
×
∼
(4)
the
path
corresponding
to
the
composite
“k
→
k
+n−1
”,
which
yields
a
∼
n−1
×
homotopy
for
pairs
of
paths
([λ
]◦[id
]◦[log]
◦[γ],
[λ
]◦[id
+n−1
]◦[γ])
—
where
n
≥
1
is
an
integer
and
[γ]
is
a
path
on
D
≤1
;
in
the
case
of
type
(4),
it
is
convenient
to
include
also
the
pair
of
paths
([λ
∼
],
[λ
×
]),
for
which
the
natural
transformation
ι
×
determines
a
homotopy.
In
addition,
it
is
natural
to
consider
the
“identity
homotopies”
associated
to
the
pairs
(5)
([γ],
[γ]),
where
[γ]
is
a
path
on
D
≤3
whose
terminal
vertex
lies
in
the
third
row
of
D
≤3
.
Thus,
if
we
take
E
log
to
be
the
set
of
ordered
pairs
of
paths
on
Γ
D
≤3
consisting
of
pairs
of
paths
of
the
above
five
types,
then
one
verifies
immediately
that
E
log
satisfies
the
conditions
(a),
(b),
(c),
(d),
(e)
given
in
§0
for
a
saturated
set.
In
particular,
the
various
homotopies
discussed
above
yield
a
family
of
homotopies
which
determines
an
observable
S
log
,
as
desired.
Moreover,
it
is
immediate
from
the
definitions
—
i.e.,
in
essence,
because
the
various
structure-orbispaces
that
appear
remain
“undisturbed”
by
the
various
manipulations
involving
arithmetic
data
that
arise
from
“ι
log,
”,
“ι
×
”
—
that
this
family
of
homotopies
is
compatible
with
the
families
of
homotopies
that
constitute
the
core
and
telecore
structures
of
(i),
(ii).
This
completes
the
proof
of
assertion
(iii).
Next,
we
consider
assertion
(iv).
Suppose
that
D
≤2
admits
a
structure
of
core
on
D
≤1
in
a
fashion
that
is
compatible
with
the
observable
S
log
of
(iii).
Then
this
core
structure
determines
a
homotopy
ζ
0
for
the
pair
of
paths
([id
],
[id
−1
]
◦
[log])
[for
∈
L];
thus,
by
composing
the
result
ζ
0
of
applying
λ
×
to
ζ
0
with
the
homotopy
ζ
1
associated
[via
S
log
]
to
the
pair
of
paths
([λ
×
]
◦
[id
−1
]
◦
[log],
[λ
∼
]
◦
[id
])
[of
type
(1)]
and
then
with
the
homotopy
ζ
2
associated
[via
S
log
]
to
the
pair
of
paths
([λ
∼
]
◦
[id
],
[λ
×
]
◦
[id
])
[of
type
(4)],
we
obtain
a
natural
transformation
ζ
1
=
ζ
2
◦
ζ
1
◦
ζ
0
:
λ
×
◦
id
→
λ
×
◦
id
—
which,
in
order
for
the
desired
compatibility
to
hold,
must
coincide
with
the
“identity
homotopy”
[of
type
(5)].
On
the
other
hand,
by
writing
out
explicitly
the
meaning
of
such
an
equality
ζ
1
=
id,
we
conclude
that
we
obtain
a
contradiction
to
Lemma
4.4.
This
completes
the
proof
of
the
first
incompatibility
of
assertion
(iv).
The
proof
of
the
second
incompatibility
of
assertion
(iv)
is
entirely
similar
[cf.
the
proof
of
Corollary
3.6,
(iv)].
This
completes
the
proof
of
assertion
(iv).
Finally,
the
total
-rigidity
portion
of
assertion
(v)
follows
immediately
from
Proposition
4.2,
(i)
[cf.
also
the
final
portion
of
Proposition
4.2,
(ii)];
the
remainder
of
assertion
(v)
follows
immediately
from
the
definitions.
Remark
4.5.1.
(i)
The
“output”
of
the
observable
S
log
of
Corollary
4.5,
(iii),
may
be
summa-
rized
intuitively
in
the
following
diagram
[cf.
Remark
3.6.1,
(i)]:
...
×
k
+1
∼
k
+1
κ
κ
...
X
+1
←
∼
←
X
+1
×
k
∼
k
κ
κ
∼
←
X
←
∼
←
X
×
k
−1
∼
k
−1
κ
κ
∼
←
X
−1
...
∼
←
X
−1
...
TOPICS
IN
ABSOLUTE
ANABELIAN
GEOMETRY
III
111
×
—
where
the
arrows
“
”
are
the
natural
surjections
[cf.
ι
×
!];
k
,
for
∈
L,
is
a
×
copy
of
“k
”
that
arises,
via
id
,
from
the
vertex
of
D
≤1
;
the
arrows
“←”
are
the
×
is
obtained
by
applying
the
log-Frobenius
inclusions
arising
from
the
fact
that
k
κ
×
functor
log
to
k
+1
[cf.
ι
log,
!];
the
“
’s”
denote
the
various
Kummer
structures
involved;
the
isomorphic
“X
’s”
correspond
to
the
coricity
of
E
[cf.
Corollary
4.5,
(i)].
Finally,
the
incompatibility
assertions
of
Corollary
4.5,
(iv),
may
be
thought
of
as
a
statement
of
the
non-existence
of
some
“universal
reference
model”
×
k
model
×
that
maps
isomorphically
to
the
various
k
’s
in
a
fashion
that
is
compatible
with
the
various
arrows
“
”,
“←”
of
the
above
diagram.
(ii)
In
words,
the
essential
content
of
Corollary
4.5
may
be
understood
as
follows
[cf.
the
“intuitive
diagram”
of
(i)]:
Although
the
operation
represented
by
the
log-Frobenius
functor
is
com-
patible
with
the
[Aut-holomorphic]
structure-orbispaces,
hence
with
the
“software”
constituted
by
the
algorithms
of
Corollary
2.7,
it
is
not
compat-
ible
with
the
additive
or
multiplicative
structures
on
the
various
arithmetic
data
involved
—
cf.
Remark
3.6.1.
That
is
to
say,
more
concretely,
if
one
starts
with
an
elliptically
admissible
Aut-
holomorphic
orbispace
X
on
which
[for
some
CAF
k]
k
×
“acts
via
the
local
linear
holomorphic
structures
of
Corollary
2.7,
(e)”
[i.e.,
X
is
equipped
with
a
Kummer
κ
structure
X
k
×
],
then
applies
log
k
to
the
universal
covering
k
∼
→
k
×
to
equip
k
∼
with
a
field
structure,
with
respect
to
which
k
∼
“acts”
on
some
isomorph
X
of
X
k
∼
←
(k
∼
)
×
k
×
κ
κ
X
∼
→
X
[where
the
“×”
of
“(k
∼
)
×
”
is
taken
with
respect
to
this
field
structure
of
k
∼
],
then
∼
although
the
“actions”
of
k
×
,
(k
∼
)
×
on
X
→
X
are
not
strictly
compatible
[i.e.,
the
diagram
does
not
commute],
they
become
“compatible”
if
one
“loosens
one’s
notion
of
compatibility”
to
the
notion
of
being
“compatible
with
the
[Aut-]holomorphic
structure”
of
the
various
objects
involved
[cf.
the
analogy
of
Remark
2.7.3].
This
state
of
affairs
may
be
expressed
formally
as
a
compatibility
between
the
various
hol
in
Definition
4.1,
(i),
co-holomorphicizations
involved
[cf.
the
definition
of
C
TH
(ii),
(iii)].
In
summary,
as
should
be
evident
from
its
statement,
Corollary
4.5
is
intended
as
an
archimedean
analogue
of
Corollary
3.6.
In
particular,
the
“general
formal
content”
of
Remarks
3.6.1,
3.6.2,
3.6.3,
3.6.5,
3.6.6,
and
3.6.7
applies
to
the
present
archimedean
situation,
as
well.
Remark
4.5.2.
By
comparison
to
the
nonarchimedean
case
treated
in
§3,
certain
—
but
not
all!
—
of
the
“arrows”
that
appear
in
the
archimedean
case
go
in
the
opposite
direction
to
the
nonarchimedean
case.
This
is
somewhat
reminiscent
of
the
“product
formula”
in
elementary
number
theory,
where,
for
instance,
positive
112
SHINICHI
MOCHIZUKI
powers
of
prime
numbers
→
0
at
nonarchimedean
primes,
but
→
∞
at
archimedean
primes.
In
the
context
of
Corollary
4.5,
perhaps
the
most
important
example
of
this
phenomenon
is
given
by
“ι
×
”.
This
leads
to
a
somewhat
different
structure
for
the
observable
S
log
of
Corollary
4.5,
(iii)
—
involving
“archimedean”
homotopies
of
arbitrarily
large
“length”
[cf.
the
“non-[γ]-portion”
of
the
pairs
of
paths
of
types
(1),
(2),
(3),
(4)
in
the
proof
of
Corollary
4.5,
(iii)]
—
from
the
structure
of
the
observable
S
log
of
Corollary
3.6,
(iii)
—
which
involves
“nonarchimedean”
paths
of
bounded
“length”
[cf.
the
“non-[γ]-portion”
of
the
pairs
of
paths
of
types
(1),
(2)
in
the
proof
of
Corollary
3.6,
(iii)].
Remark
4.5.3.
(i)
By
replacing
“λ
×pf
”
by
“λ
∼
”,
“ι
×
=
ι
×
:
λ
×
→
λ
×pf
”
by
“ι
×
=
ι
×
:
λ
∼
→
λ
×
”,
and
“Corollary
1.10”
by
“Corollary
2.7”,
[and
making
various
other
suitable
revisions]
one
obtains
an
essentially
straightfor-
ward
“Aut-holomorphic
translation”
of
the
bi-anabelian
incompatibility
result
given
in
Corollary
3.7.
We
leave
the
routine
details
to
the
reader.
(ii)
The
“general
formal
content”
of
Remarks
3.7.1,
3.7.2,
3.7.3,
3.7.4,
3.7.5,
3.7.7,
and
3.7.8
applies
to
the
archimedean
analogue
of
Corollary
3.7
discussed
in
(i)
—
cf.
also
the
analogy
of
Remark
2.7.3;
the
discussion
in
Remark
2.7.4
of
“fixed
reference
models”
in
the
context
of
the
definition
of
the
notion
of
a
“holomorphic
structure”.
(iii)
With
regard
to
the
discussion
in
Remark
3.7.4
of
“functorially
trivial
mod-
els”
[i.e.,
models
that
“arise
from
Π”
without
essential
use
of
Π,
hence
are
equipped
with
trivial
functorial
actions
of
Π],
we
note
that
although
“the
Galois
group
Π”
does
not
appear
in
the
present
archimedean
context,
the
“functorial
detachment”
of
such
“functorially
trivial
models”
means,
for
instance,
that
if
one
regards
some
model
X
model
as
“arising”
from
an
elliptically
admissible
Aut-holomorphic
orbis-
pace
X
in
a
“trivial
fashion”,
then
when
one
applies
the
“elliptic
cuspidalization”
portion
of
the
algorithm
of
Corollary
2.7,
(b),
the
various
coverings
of
X
involved
in
this
elliptic
cuspidalization
algorithm
functorially
induce
trivial
coverings
of
X
model
,
hence
do
not
give
rise
to
a
functorial
isomorphism
of
the
respective
“base
fields”
[cf.
Remark
2.7.3]
of
X,
X
model
.
(iv)
With
regard
to
the
discussion
in
Remark
3.7.5,
one
may
give
an
archimedean
analogue
of
the
“pathological
versions
of
the
Kummer
map”
given
in
Remark
3.7.5,
∼
(ii),
by
composing
the
k-Kummer
structure
[cf.
Definition
4.1,
(i)]
“κ
k
:
k
→
A
X
ell
”,
restricted,
say,
to
k
×
,
with
the
[non-additive!]
automorphism
of
∼
k
×
→
O
k
×
×
R
>0
that
acts
as
the
identity
on
O
k
×
and
is
given
by
raising
to
the
λ-th
power
[for
some
λ
∈
R
>0
]
on
R
>0
.
TOPICS
IN
ABSOLUTE
ANABELIAN
GEOMETRY
III
113
Section
5:
Global
Log-Frobenius
Compatibility
In
the
present
§5,
we
globalize
the
theory
of
§3,
§4.
This
globalization
allows
one
to
construct
canonical
rigid
compacta
—
i.e.,
canonical
integral
structures
—
that
enable
one
to
consider
[“pana-”]localizations
of
global
arithmetic
line
bundles
[cf.
Corollary
5.5]
without
obliterating
the
“volume-theoretic”
information
inherent
in
the
theory
of
global
arithmetic
degrees,
and
in
a
fashion
that
is
compatible
with
the
operation
of
“mono-analyticization”
[cf.
Corollary
5.10]
—
i.e.,
the
operation
of
“disabling
the
rigidity”
of
one
of
the
“two
combinatorial
dimensions”
of
a
ring
[cf.
Remark
5.6.1].
The
resulting
theory
is
reminiscent,
in
certain
formal
respects,
of
the
p-adic
Teichmüller
theory
of
[Mzk1],
[Mzk4]
[cf.
Remark
5.10.3].
Definition
5.1.
(i)
Let
F
be
a
number
field.
Then
we
shall
write
V(F
)
for
the
set
of
[archimedean
def
{
F
},
where
the
and
nonarchimedean]
valuations
of
F
,
and
V
(F
)
=
V(F
)
symbol
“
F
”
is
to
be
thought
of
as
representing
the
global
field
F
,
or,
alternatively,
the
generic
prime
of
F
.
If
F
is
an
algebraic
closure
of
F
,
then
we
shall
write
def
V(F
/F
)
=
lim
←−
V(K);
K
V
(F
/F
)
=
lim
←−
V
(K)
def
K
[where
K
ranges
over
the
finite
extensions
of
F
in
F
]
for
the
inverse
limits
relative
to
the
evident
systems
of
morphisms.
The
inverse
system
of
“
K
’s”
determines
a
unique
global
element
F
∈
V
(F
/F
);
the
other
elements
of
V
(F
/F
)
lie
in
the
image
of
the
natural
injection
V(F
/F
)
→
V
(F
/F
)
and
will
be
called
local;
moreover,
we
have
a
natural
decomposition
V(F
/F
)
=
V(F
/F
)
arc
V(F
/F
)
non
into
archimedean
and
nonarchimedean
local
elements.
There
is
a
natural
contin-
uous
action
of
Gal(F
/F
)
on
the
pro-sets
V(F
/F
),
V
(F
/F
).
For
K
⊆
F
a
fi-
nite
extension
of
F
,
V(K),
V
(K)
may
be
identified,
respectively,
with
the
sets
of
Gal(F
/K)
(⊆
Gal(F
/F
))-orbits
V(F
/F
)/Gal(F
/K),
V
(F
/F
)/Gal(F
/K)
of
V(F
/F
),
V
(F
/F
).
(ii)
Let
X
be
an
elliptically
admissible
[cf.
[Mzk21],
Definition
3.1]
hyperbolic
orbicurve
over
a
totally
imaginary
number
field
F
[so
X
is
also
of
strictly
Belyi
type
—
cf.
Remark
2.8.3].
Write
Π
X
for
the
étale
fundamental
group
of
X
[for
def
some
choice
of
basepoint];
Π
X
G
F
=
Gal(F
/F
)
for
the
natural
surjection
onto
the
absolute
Galois
group
G
F
of
F
[for
some
choice
of
algebraic
closure
F
of
F
];
Δ
X
⊆
Π
X
for
the
kernel
of
this
surjection
[which
may
be
characterized
“group-
theoretically”
as
the
maximal
topologically
finite
generated
closed
normal
subgroup
of
Π
X
—
cf.,
e.g.,
[Mzk9],
Lemma
1.1.4,
(i)].
Write
F
mod
⊆
F
for
the
“field
of
moduli
of
X”,
i.e.,
the
subfield
of
F
determined
by
the
[open]
image
def
of
Aut(X
F
)
[i.e.,
the
group
of
automorphisms
of
the
scheme
X
F
=
X
×
F
F
]
in
114
SHINICHI
MOCHIZUKI
Aut(F
)
=
Gal(F
/Q)
(⊇
G
F
);
Aut(X),
Aut(F
)
for
the
respective
automorphism
groups
of
the
schemes
X,
Spec(F
).
For
simplicity,
we
also
make
the
following
assumption
on
X:
F
is
Galois
over
F
mod
;
the
natural
homomorphism
Aut(X)
→
Aut(F
)
surjects
onto
Gal(F/F
mod
)
(⊆
Aut(F
));
we
have
a
natural
isomorphism
∼
Aut(X/F
)
→
Aut(X
F
/F
)
between
the
group
of
F
-automorphisms
of
X
and
the
group
of
F
-automor-
phisms
of
X
F
.
This
assumption
on
X
implies
that
we
have
a
natural
isomorphism
∼
Aut(X)
×
Gal(F/F
mod
)
Gal(F
/F
mod
)
→
Aut(X
F
)
[induced
by
the
fiber
product
structure
X
F
=
X
×
F
F
],
and
hence
that
the
natural
exact
sequence
1
→
Aut(X
F
/F
)
→
Aut(X
F
)
→
Gal(F
/F
mod
)
→
1
admits
a
natural
surjection
onto
the
natural
exact
sequence
1
→
Aut(X/F
)
→
Aut(X)
→
Gal(F/F
mod
)
→
1
[induced
by
the
projection
Aut(X)
×
Gal(F/F
mod
)
Gal(F
/F
mod
)
Aut(X)
to
the
first
factor].
Note
that
by
the
functoriality
of
the
algorithms
of
Theorem
1.9,
it
fol-
∼
lows
that
there
is
a
natural
isomorphism
Aut(X)
→
Out(Π
X
)
that
is
compatible
with
the
natural
morphisms
Aut(X)
Gal(F/F
mod
)
→
Aut(F
)
∼
=
Out(G
F
)
[cf.,
e.g.,
[Mzk15],
Theorem
3.1],
Out(Π
X
)
→
Out(G
F
);
in
particular,
one
may
functorially
construct
the
image
G
F
mod
→
Aut(G
F
)
as
the
inverse
image
[i.e.,
via
the
natural
projection
Aut(G
F
)
→
Out(G
F
)]
of
the
image
of
Out(Π
X
)
→
Out(G
F
).
Next,
ob-
×
serve
that
one
may
functorially
construct
“F
”
from
Π
X
as
the
field
“k
NF
{0}
(
∼
=
k
NF
)”
constructed
in
Theorem
1.9,
(e)
[cf.
also
Remark
1.10.1,
(i)];
denote
this
field
×
constructed
from
Π
X
by
k
NF
(Π
X
);
we
shall
also
use
the
notation
k
NF
(Π
X
)
for
the
group
of
nonzero
elements
of
this
field.
In
particular,
by
considering
[cf.
Corollary
2.8]
valuations
on
the
field
k
NF
(Π
X
)
[where
each
valuation
is
valued
in
the
“copy
of
×
R”
given
by
completing
the
group
“k
NF
”
with
respect
to
the
“order
topology”
de-
termined
by
the
valuation],
one
may
functorially
construct
“V
(F
/F
)”,
“V(F
/F
)”
from
Π
X
;
denote
the
resulting
pro-sets
constructed
in
this
way
by
V
(Π
X
),
V(Π
X
)
and
the
completion
of
k
NF
(Π
X
)
at
v
∈
V(Π
X
)
by
k
NF
(Π
X
,
v).
For
v
∈
V(F
/F
)
non
,
write
Π
X,v
⊆
Π
X
for
the
decomposition
group
of
v
[i.e.,
the
closed
subgroup
of
elements
of
Π
X
that
fix
v];
for
v
∈
V(F
/F
)
arc
,
write
X
ell,v
for
the
Aut-holomorphic
orbispace
“X
v
”
[associated
to
X
at
v]
of
Corollary
2.8,
∼
δ
ell,v
:
Δ
X
→
π
1
(X
ell,v
)
∧
for
the
natural
outer
isomorphism
of
Δ
X
with
the
profinite
completion
[denoted
by
the
superscript
“∧”]
of
the
topological
fundamental
group
of
X
ell,v
,
and
κ
ell,v
:
k
NF
(Π
X
)
→
A
X
ell,v
TOPICS
IN
ABSOLUTE
ANABELIAN
GEOMETRY
III
115
for
the
natural
inclusion
of
fields
[i.e.,
arising
from
the
isomorphism
of
topological
fields
of
Corollary
2.9,
(b)].
When
we
wish
to
regard
X
ell,v
as
an
object
constructed
from
Π
X
[cf.
Corollary
2.8],
we
shall
use
the
notation
X(Π
X
,
v)
[where
we
regard
v
as
an
element
of
V(Π
X
)
arc
].
Finally,
we
observe
that
Aut(Π
X
)
acts
naturally
on
all
of
these
objects
constructed
from
Π
X
.
In
particular,
we
have
a
natural
bijection
def
∼
V
(Π
X
)/Aut(Π
X
)
→
V
(F
mod
).
For
v
∈
V(F
mod
),
write
d
mod
=
[F
v
F
:
(F
mod
)
v
]
v
for
the
degree
of
the
completion
of
F
at
any
v
F
∈
V(F
)
that
divides
v
over
the
completion
(F
mod
)
v
of
F
mod
at
v.
(iii)
Write
EA
for
the
category
whose
objects
are
profinite
groups
isomorphic
to
Π
X
for
some
X
as
in
(ii),
and
whose
morphisms
are
open
injections
of
profinite
groups
that
in-
duce
isomorphisms
between
the
respective
maximal
topologically
finitely
generated
closed
normal
subgroups
[i.e.,
the
respective
“Δ
X
”].
We
shall
refer
to
as
a
global
Galois-theater
any
collection
of
data
V
=
(Π
V
,
{Π
v
}
v∈V
non
,
{(X
v
,
δ
v
,
κ
v
)}
v∈V
arc
)
def
—
where
Π
∈
Ob(EA
);
we
shall
refer
to
Π
as
the
global
Galois
group
of
the
Galois-
theater;
we
write
Δ
⊆
Π
for
the
maximal
topologically
finitely
generated
closed
normal
subgroup
of
Π;
V
is
a
pro-set
equipped
with
a
continuous
action
by
Π
that
non
arc
non
arc
def
decomposes
into
a
disjoint
union
V
=
{
V
}
V
V
⊇
V
=
V
V
;
non
arc
for
v
∈
V
,
Π
v
⊆
Π
is
the
closed
subgroup
of
elements
that
fix
v;
for
v
∈
V
,
∼
X
v
is
an
Aut-holomorphic
orbispace,
δ
v
:
Δ
→
π
1
(X
v
)
∧
is
an
outer
isomorphism
of
profinite
groups,
and
κ
v
:
k
NF
(Π)
→
A
X
v
is
an
inclusion
of
fields
—
such
that
there
exists
a(n)
[unique!
—
cf.
Remark
5.1.1
below]
isomorphism
of
pro-sets
∼
ψ
V
:
V
(Π)
→
V
—
which
we
shall
refer
to
as
a
reference
isomorphism
for
V
—
that
satisfies
the
fol-
∼
lowing
conditions:
(a)
ψ
V
is
Π-equivariant
and
maps
k
NF
(Π)
→
V
,
V
(Π)
non
→
non
∼
arc
arc
V
,
V
(Π)
arc
→
V
;
(b)
for
V
(Π)
arc
v
ell
→
v
∈
V
,
there
exists
a(n)
∼
[unique!
—
cf.
Remark
5.1.1
below]
isomorphism
ψ
v
:
X(Π,
v
ell
)
→
X
v
of
Aut-
holomorphic
spaces
that
is
compatible
with
δ
ell,v
ell
,
δ
v
,
as
well
as
with
κ
ell,v
ell
,
κ
v
.
A
morphism
of
global
Galois-theaters
φ
:
(Π
1
V
1
,{(Π
1
)
v
1
},
{((X
1
)
v
1
,
δ
v
1
,
κ
v
1
)})
→
(Π
2
V
2
,
{(Π
2
)
v
2
},
{((X
2
)
v
2
,
δ
v
2
,
κ
v
2
)})
is
defined
to
consist
of
a
morphism
φ
Π
:
Π
1
→
Π
2
of
EA
and
a(n)
[uniquely
∼
determined
—
cf.
Remark
5.1.1
below]
isomorphism
of
pro-sets
φ
V
:
V
1
→
V
2
that
satisfy
the
following
conditions:
(a)
φ
Π
,
φ
V
are
compatible
with
the
actions
non
∼
non
arc
∼
arc
of
Π
1
,
Π
2
on
V
1
,
V
2
,
and
map
V
1
→
V
2
,
V
1
→
V
2
,
V
1
→
V
2
;
(b)
arc
arc
for
V
1
v
1
→
v
2
∈
V
2
,
there
exists
a
[unique!
—
cf.
Remark
5.1.1
below]
∼
isomorphism
φ
v
:
X
v
1
→
X
v
2
of
Aut-holomorphic
spaces
that
is
compatible
with
116
SHINICHI
MOCHIZUKI
non
δ
v
1
,
δ
v
2
,
as
well
as
with
κ
v
1
,
κ
v
2
.
[Here,
we
note
that
(a)
implies
that
for
V
1
non
v
1
→
v
2
∈
V
2
,
φ
Π
induces
an
open
injection
φ
v
:
Π
v
1
→
Π
v
2
.]
(iv)
In
the
notation
of
(iii),
we
shall
refer
to
as
a
panalocal
Galois-theater
any
collection
of
data
def
V
=
(V
,
{Π
v
}
v∈V
non
,
{X
v
}
v∈V
arc
)
—
where
V
is
a
set
that
decomposes
as
a
disjoint
union
V
=
{
V
}
V
non
V
arc
def
⊇
V
=
V
non
V
arc
;
for
v
∈
V
non
,
Π
v
∈
Ob(Orb(TG))
[cf.
§0;
Definition
3.1,
(iii)];
for
v
∈
V
arc
,
X
v
∈
Ob(Orb(EA))
[cf.
Definition
4.1,
(iii)]
—
such
that
there
exists
a
Π
∈
Ob(EA
)
and
an
isomorphism
of
sets
∼
ψ
V
:
V
(Π)/Aut(Π)
→
V
—
which
we
shall
refer
to
as
a
reference
isomorphism
for
V
—
that
satisfies
the
following
conditions:
(a)
the
composite
of
ψ
V
with
the
quotient
map
V
(Π)
V
(Π)/Aut(Π)
maps
k
NF
(Π)
→
V
,
V(Π)
non
V
non
,
V(Π)
arc
V
arc
;
(b)
for
each
v
∈
V
non
,
Π
v
is
isomorphic
to
the
object
of
Orb(TG)
determined
by
“the
decomposition
group
Π
v
⊆
Π
of
v,
considered
up
to
automorphisms
of
Π
v
,
as
v
∈
V(Π)
ranges
over
the
elements
lying
over
v”;
(c)
for
each
v
∈
V
arc
,
X
v
is
isomorphic
to
the
object
of
Orb(EA)
determined
by
“the
Aut-holomorphic
orbispace
X(Π,
v),
considered
up
to
automorphisms
of
X(Π,
v),
as
v
∈
V(Π)
ranges
over
the
elements
lying
over
v”.
A
morphism
of
panalocal
Galois-theaters
φ
:
(V
1
,
{(Π
1
)
v
1
},
{(X
1
)
v
1
})
→
(V
2
,
{(Π
2
)
v
2
},
{(X
2
)
v
2
})
∼
is
defined
to
consist
of
a
bijection
of
sets
φ
V
:
V
1
→
V
2
that
induces
bijections
∼
∼
V
1
non
→
V
2
non
,
V
1
arc
→
V
2
arc
,
together
with
open
injections
of
[orbi-]profinite
groups
(Π
1
)
v
1
→
(Π
2
)
v
2
[where
V
1
non
v
1
→
v
2
∈
V
2
non
;
we
recall
that,
in
the
notation
of
(ii),
“F/F
mod
”
is
Galois],
and
isomorphisms
of
[orbi-]Aut-holomorphic
orbis-
∼
paces
(X
1
)
v
1
→
(X
2
)
v
2
[where
V
1
arc
v
1
→
v
2
∈
V
2
arc
].
[Here,
we
observe
that
∼
the
existence
of
the
isomorphisms
“(X
1
)
v
1
→
(X
2
)
v
2
”
implies
—
by
considering
Euler
characteristics
[cf.
also
[Mzk20],
Theorem
2.6,
(v)]
—
that
the
open
injec-
∼
tions
“(Π
1
)
v
1
→
(Π
2
)
v
2
”
induce
isomorphisms
“Δ
1
→
Δ
2
”
between
the
respective
geometric
fundamental
groups.]
Write
Th
(respectively,
Th
)
for
the
category
of
global
(respectively,
panalocal)
Galois-theaters
and
morphisms
of
global
(respec-
tively,
panalocal)
Galois-theaters.
Thus,
it
follows
immediately
from
the
definitions
that
we
obtain
a
natural
“panalocalization
functor”
Th
→
Th
—
which
is
essentially
surjective.
(v)
Let
T
∈
{TF,
TM,
TLG}
[cf.
the
notation
of
Definition
3.1,
(i)].
If
T
=
TF,
def
def
then
let
T
=
T;
if
T
=
TF,
then
let
T
=
TLG;
if
T
=
T,
then
a
superscript
“T
”
will
be
used
to
denote
the
operation
of
groupification
of
a
monoid
[i.e.,
“gp”];
if
T
=
T,
then
a
superscript
“T
”
will
be
used
to
denote
the
“identity
operation”
[i.e.,
may
be
ignored].
If
Π
∈
Ob(EA
),
then
let
us
write
M
T
(Π)
for
the
object
of
T
,
equipped
with
a
continuous
action
by
Π,
determined
by
k
NF
(Π)
[if
T
=
TF],
TOPICS
IN
ABSOLUTE
ANABELIAN
GEOMETRY
III
117
×
k
NF
(Π)
[if
T
=
TLG],
equipped
with
the
discrete
topology;
if
v
∈
V(Π),
then
let
us
write
M
T
(Π,
v)
for
the
object
of
T,
equipped
with
a
continuous
action
by
the
×
decomposition
group
Π
v
⊆
Π
of
v,
determined
by
k
NF
(Π,
v)
[if
T
=
TF],
k
NF
(Π,
v)
[if
T
=
TM].
A
global
T-pair
is
defined
to
be
a
collection
[if
T
=
TLG],
O
k
NF
(Π,v)
of
data
κ
M
=
(V
,
M
,
{ρ
v
}
v∈V
,
{(Π
v
M
v
)}
v∈V
non
,
{(X
v
M
v
)}
v∈V
arc
)
def
—
where
V
=
(Π
V
,
{Π
v
}
v∈V
non
,
{(X
v
,
δ
v
,
κ
v
)}
v∈V
arc
)
non
arc
V
;
M
∈
Ob(T
),
which
we
shall
refer
is
a
global
Galois-theater;
V
=
V
to
as
the
global
arithmetic
data
of
M
,
is
equipped
with
a
continuous
action
by
non
Π;
for
each
v
∈
V
,
(Π
v
M
v
)
is
an
MLF-Galois
T-pair
with
Galois
group
arc
κ
given
by
Π
v
;
for
each
v
∈
V
,
(X
v
M
v
)
is
an
Aut-holomorphic
T-pair
with
structure-orbispace
given
by
X
v
;
for
each
v
∈
V
,
ρ
v
:
M
→
M
v
T
is
a
[“re-
striction”]
morphism
in
T
—
such
that,
relative
to
some
reference
isomorphism
∼
ψ
V
:
V
(Π)
→
V
for
V
as
in
(iii),
there
exist
isomorphisms
[in
T
,
T,
respec-
tively]
∼
∼
ψ
:
M
T
(Π)
→
M
;
{ψ
v
:
M
T
(Π,
v)
→
M
v
}
v∈V
—
which
we
shall
refer
to
as
reference
isomorphisms
for
M
—
that
satisfy
the
non
,
ψ
v
is
Π
v
-equivariant;
following
conditions:
(a)
ψ
is
Π-equivariant;
(b)
for
v
∈
V
arc
κ
(c)
for
v
∈
V
,
the
composite
of
ψ
v
with
the
Kummer
structure
of
(X
v
M
v
)
is
compatible
with
κ
v
;
(d)
ψ
,
{ψ
v
}
v∈V
are
compatible
with
the
{ρ
v
}
v∈V
,
relative
to
the
natural
restriction
morphisms
ρ
v
(Π)
:
M
T
(Π)
→
M
T
(Π,
v)
T
.
In
this
situation,
if
T
=
TF,
then
we
shall
refer
to
the
profinite
Π-module
μ
(M
)
=
Hom(Q/Z,
M
)
Z
def
as
the
cyclotome
associated
to
this
global
T-pair
and
[which
is
isomorphic
to
Z]
def
(M
)
⊗
Q/Z.
A
morphism
of
global
T-pairs
write
μ
Q/Z
(M
)
=
μ
Z
κ
φ
:
(V
1
,M
1
,
{ρ
v
1
},
{((Π
1
)
v
1
(M
1
)
v
1
)},
{((X
1
)
v
1
(M
1
)
v
1
)})
κ
→
(V
2
,
M
2
,
{ρ
v
2
},
{((Π
2
)
v
2
(M
2
)
v
2
)},
{((X
2
)
v
2
(M
2
)
v
2
)})
is
defined
to
consist
of
a
morphism
of
global
Galois-theaters
φ
V
:
V
1
→
V
2
,
∼
together
with
an
isomorphism
φ
:
M
1
→
M
2
of
T
,
and
isomorphisms
φ
v
1
:
∼
v
1
→
v
2
∈
V
2
]
in
T,
that
satisfy
the
following
(M
1
)
v
1
→
(M
2
)
v
2
[where
V
1
compatibility
conditions:
(a)
φ
is
equivariant
with
respect
to
the
open
injection
non
Π
1
→
Π
2
arising
from
φ
V
;
(b)
for
v
1
∈
V
1
,
the
isomorphism
φ
v
1
is
compatible
with
the
actions
of
(Π
1
)
v
1
,
(Π
2
)
v
2
,
relative
to
the
open
injection
(Π
1
)
v
1
→
(Π
2
)
v
2
arc
induced
by
φ
V
;
(c)
for
v
1
∈
V
1
,
the
isomorphism
φ
v
1
is
compatible
with
the
κ
κ
Kummer
structures
of
((X
1
)
v
1
(M
1
)
v
1
),
((X
2
)
v
2
(M
2
)
v
2
),
relative
to
the
iso-
∼
morphism
(X
1
)
v
1
→
(X
2
)
v
2
induced
by
φ
V
;
(d)
φ
,
{φ
v
1
}
v
1
∈V
1
are
compatible
with
the
{ρ
v
1
}
v
1
∈V
1
,
{ρ
v
2
}
v
2
∈V
2
.
118
SHINICHI
MOCHIZUKI
(vi)
In
the
notation
of
(v),
a
panalocal
T-pair
is
defined
to
be
a
collection
of
data
κ
M
=
(V
,
{(Π
v
M
v
)}
v∈V
non
,
{(X
v
M
v
)}
v∈V
arc
)
def
—
where
V
=
(V
,
{Π
v
}
v∈V
non
,
{X
v
}
v∈V
arc
)
is
a
panalocal
Galois-theater;
for
each
v
∈
V
non
,
(Π
v
M
v
)
is
a(n)
[strictly
speaking,
“orbi-”]MLF-Galois
T-pair
with
κ
Galois
group
given
by
Π
v
;
for
each
v
∈
V
arc
,
(X
v
M
v
)
is
a(n)
[strictly
speaking,
“orbi-”]Aut-holomorphic
T-pair
with
structure-orbispace
given
by
X
v
.
A
morphism
of
panalocal
T-pairs
κ
φ
:
(V
1
,{((Π
1
)
v
1
(M
1
)
v
1
)},
{((X
1
)
v
1
(M
1
)
v
1
)})
κ
→
(V
2
,
{((Π
2
)
v
2
(M
2
)
v
2
)},
{((X
2
)
v
2
(M
2
)
v
2
)})
is
defined
to
consist
of
a
morphism
of
panalocal
Galois-theaters
φ
V
:
V
1
→
V
2
,
together
with
compatible
T-isomorphisms
of
[orbi-]MLF-Galois
T-pairs
φ
v
1
:
v
1
→
v
2
∈
V
2
non
]
and
((Π
1
)
v
1
(M
1
)
v
1
)
→
((Π
2
)
v
2
(M
2
)
v
2
)
[where
V
1
non
κ
κ
[orbi-]Aut-holomorphic
T-pairs
φ
v
1
:
((X
1
)
v
1
(M
1
)
v
1
)
→
((X
2
)
v
2
(M
2
)
v
2
)
v
1
→
v
2
∈
V
2
arc
].
Write
Th
[where
V
1
arc
T
(respectively,
Th
T
)
for
the
category
of
global
(respectively,
panalocal)
T-pairs
and
morphisms
of
global
(respectively,
panalocal)
T-pairs.
Thus,
it
follows
immediately
from
the
definitions
that
we
ob-
tain
a
natural
“panalocalization
functor”
Th
T
→
Th
T
—
lying
over
the
functor
Th
→
Th
of
(iv)
—
which
is
essentially
surjective.
Moreover,
we
have
compatible
natural
functors
Th
→
EA
,
Th
T
→
EA
,
as
well
as
natural
functors
Th
TF
→
Th
TM
;
Th
TM
→
Th
TLG
;
Th
TF
→
Th
TM
;
Th
TM
→
Th
TLG
[cf.
Definition
3.1,
(iii);
Definition
4.1,
(iii)].
Remark
5.1.1.
Note
that
the
reference
isomorphism
ψ
V
of
Definition
5.1,
(iii),
is
uniquely
determined
by
the
conditions
stated.
Indeed,
for
nonarchimedean
elements,
non
this
follows
by
considering
the
stabilizers
in
Π
of
elements
of
V
,
together
with
the
well-known
fact
that
a
nonarchimedean
prime
of
F
[cf.
the
notation
of
Definition
5.1,
(ii)]
is
uniquely
determined
by
any
open
subgroup
of
its
decomposition
group
in
G
F
[cf.,
e.g.,
[NSW],
Corollary
12.1.3];
for
archimedean
elements,
this
follows
arc
by
considering
the
topology
induced
on
k
NF
(Π)
by
A
X
v
via
“κ
v
”
for
v
∈
V
.
arc
Moreover,
for
v
∈
V
,
the
isomorphism
ψ
v
of
Definition
5.1,
(iii),
is
uniquely
determined
by
the
condition
of
compatibility
with
δ
ell,v
ell
,
δ
v
.
Indeed,
by
Corollary
2.3,
(i)
[cf.
also
[Mzk14],
Lemma
1.3,
(iii)],
this
follows
from
the
well-known
fact
that
any
automorphism
of
a
hyperbolic
orbicurve
that
induces
the
identity
outer
automorphism
of
the
profinite
fundamental
group
of
the
orbicurve
is
itself
the
identity
automorphism.
Similar
uniqueness
statements
[with
similar
proofs]
hold
for
the
morphisms
φ
V
,
φ
v
of
Definition
5.1,
(iii).
Corollary
5.2.
(First
Properties
of
Galois-theaters
and
Pairs)
Let
T
∈
{TF,
TM}.
We
shall
apply
a
subscript
“TM”
to
[global
or
local]
arithmetic
data
of
TOPICS
IN
ABSOLUTE
ANABELIAN
GEOMETRY
III
119
“T-pairs”
to
denote
the
result
of
applying
the
natural
functor
whose
codomain
is
the
corresponding
category
of
“TM-pairs”
[i.e.,
the
identity
functor
if
T
=
TM
—
cf.
Proposition
3.2];
we
shall
also
use
the
subscript
“TLG”
in
a
similar
way.
(i)
Write
An
[Th
]
for
the
category
whose
objects
are
data
of
the
form
V
(Π)
=
(Π
V
(Π),
{Π
v
}
v∈V(Π)
non
,
{(X(Π,
v),
δ
ell,v
,
κ
ell,v
)}
v∈V(Π)
arc
)
def
[cf.
the
notation
of
Definition
5.1,
(i)]
for
Π
∈
Ob(EA
)
and
whose
morphisms
are
the
morphisms
induced
by
morphisms
of
EA
.
Then
we
have
natural
functors
EA
→
An
[Th
]
→
Th
→
EA
—
where
the
first
arrow
is
the
functor
obtained
by
assigning
Ob(EA
)
Π
→
V
(Π);
the
second
arrow
is
the
functor
obtained
by
forgetting
the
way
in
which
the
global
Galois-theater
data
V
(Π)
arose
from
Π;
the
third
arrow
is
the
natu-
ral
functor
of
Definition
5.1,
(vi);
the
composite
EA
→
EA
of
these
arrows
is
naturally
isomorphic
to
the
identity
functor
—
all
of
which
are
equivalences
of
categories.
(ii)
Let
κ
(V
,
M
,
{ρ
v
}
v∈V
,
{(Π
v
M
v
)}
v∈V
non
,
{(X
v
M
v
)}
v∈V
arc
)
be
a
global
T-pair
[as
in
Definition
5.1,
(v)].
Then
there
is
a
unique
[hence,
in
par-
ticular,
there
exists
a
functorial
—
relative
to
Th
T
—
algorithm
for
constructing
the]
isomorphism
∼
(M
TM
)
→
μ
(Π)
μ
Z
Z
[cf.
Theorem
1.9,
(b);
Remark
1.10.1,
(ii)]
of
Π-modules
that
is
compatible
—
relative
to
the
restriction
morphisms
{ρ
v
}
v∈V
non
—
with
the
isomorphisms
non
∼
((M
v
)
TM
)
→
μ
(Π
v
),
for
v
∈
V
,
obtained
by
composing
the
isomorphisms
of
μ
Z
Z
Corollary
1.10,
(c);
Remark
3.2.1.
(iii)
In
the
notation
of
(ii),
there
exists
a
functorial
[i.e.,
relative
to
Th
T
]
algorithm
for
constructing
the
Kummer
map
∼
∼
∼
gp
1
1
M
TM
→
(M
TM
)
→
M
TLG
→
lim
(M
TM
))
→
lim
(Π))
−→
H
(J,
μ
−→
H
(J,
μ
Z
Z
J
J
—
where
“J”
ranges
over
the
open
subgroups
of
Π.
In
particular,
the
reference
isomorphisms
ψ
,
{ψ
v
}
of
Definition
5.1,
(v),
are
uniquely
determined
by
the
conditions
stated
in
Definition
5.1,
(v);
in
a
similar
vein,
the
isomorphisms
φ
,
{φ
v
}
that
appear
in
the
definition
of
a
“morphism
φ
of
global
T-pairs”
in
Definition
5.1,
(v),
are
uniquely
determined
by
φ
V
.
(iv)
Write
An
[Th
T
]
for
the
category
whose
objects
are
data
of
the
form
M
T
(Π)
=
(V
(Π),M
T
(Π),
{ρ
v
(Π)}
v∈V(Π)
,
κ
{(Π
v
M
T
(Π,
v))}
v∈V(Π)
non
,
{(X(Π,
v)
M
T
(Π,
v))}
v∈V(Π)
arc
)
def
120
SHINICHI
MOCHIZUKI
[cf.
the
notation
of
Definition
5.1,
(v)]
for
Π
∈
Ob(EA
)
and
whose
morphisms
are
the
morphisms
induced
by
morphisms
of
EA
.
Then
we
have
natural
functors
EA
→
An
[Th
T
]
→
Th
T
→
EA
—
where
the
first
arrow
is
the
functor
obtained
by
assigning
Ob(EA
Π
→
T
)
(Π);
the
second
arrow
is
the
functor
obtained
by
forgetting
the
way
in
which
M
T
the
global
T-pair
data
M
T
(Π)
arose
from
Π;
the
third
arrow
is
the
natural
functor
of
Definition
5.1,
(vi);
the
composite
EA
→
EA
of
these
arrows
is
naturally
isomorphic
to
the
identity
functor
—
all
of
which
are
equivalences
of
categories
that
are
[1-]compatible
[in
the
evident
sense]
with
the
functors
of
(i).
(v)
Write
An
[Th
]
for
the
category
whose
objects
are
data
of
the
form
V
(Π)
=
(Π,
{V
(Π)}
)
def
—
where
V
(Π)
is
as
in
(i);
we
use
the
notation
“{−}
”
to
denote
the
data
obtained
by
applying
the
panalocalization
functor
Th
→
Th
of
Definition
5.1,
(iv)
—
for
Π
∈
Ob(EA
)
and
whose
morphisms
are
the
morphisms
induced
by
morphisms
of
EA
.
Then
we
have
natural
functors
EA
→
An
[Th
]
→
Th
—
where
the
first
arrow
is
the
functor
obtained
by
assigning
Ob(EA
)
Π
→
V
(Π);
the
second
arrow
is
the
functor
obtained
by
forgetting
the
way
in
which
the
panalocal
Galois-theater
data
{V
(Π)}
arose
from
Π.
Here,
the
first
arrow
EA
→
An
[Th
]
is
an
equivalence
of
categories.
(vi)
Write
An
[Th
T
]
for
the
category
whose
objects
are
data
of
the
form
M
T
(Π)
=
(Π,
{M
T
(Π)}
)
def
—
where
M
T
(Π)
is
as
in
(iv);
we
use
the
notation
“{−}
”
to
denote
the
data
obtained
by
applying
the
panalocalization
functor
Th
→
Th
of
Definition
5.1,
(vi)
—
for
Π
∈
Ob(EA
)
and
whose
morphisms
are
the
morphisms
induced
by
morphisms
of
EA
.
Then
we
have
natural
functors
EA
→
An
[Th
T
]
→
Th
T
—
where
the
first
arrow
is
the
functor
obtained
by
assigning
Ob(EA
)
Π
→
M
T
(Π);
the
second
arrow
is
the
functor
obtained
by
forgetting
the
way
in
which
arose
from
Π.
Here,
the
first
arrow
EA
→
the
panalocal
T-pair
data
{M
T
(Π)}
An
[Th
T
]
is
an
equivalence
of
categories.
(vii)
By
replacing,
in
the
definition
of
the
objects
of
Th
T
[cf.
Definition
5.1,
(iv)],
the
data
in
Orb(TG)
(respectively,
Orb(EA))
labeled
by
a(n)
nonar-
chimedean
(respectively,
archimedean)
valuation
by
[the
result
of
applying
(−)
T
to]
the
data
that
constitutes
the
corresponding
object
of
Orb(Anab)
[cf.
Definition
3.1,
(vi)]
(respectively,
Orb(LinHol)
[cf.
Definition
4.1,
(v)]),
we
obtain
a
category
An
[Th
T
]
TOPICS
IN
ABSOLUTE
ANABELIAN
GEOMETRY
III
121
—
i.e.,
whose
morphisms
are
the
morphisms
induced
by
morphisms
of
Th
—
together
with
natural
functors
Th
→
An
[Th
T
]
→
Th
T
→
Th
—
where
the
first
arrow
is
the
functor
arising
from
the
definition
of
An
[Th
T
];
the
second
arrow
is
the
“forgetful
functor”
[cf.
the
“forgetful
functors”
of
assertion
(ii)
of
Corollaries
3.6,
4.5];
the
third
arrow
is
the
natural
functor
[cf.
Definition
5.1,
(vi)];
the
composite
Th
→
Th
of
these
arrows
is
naturally
isomorphic
to
the
identity
functor
—
all
of
which
are
equivalences
of
categories.
Proof.
In
light
of
Remark
5.1.1,
assertion
(i)
is
immediate
from
the
definitions
and
the
results
of
§1,
§2
[cf.,
especially,
Theorem
1.9;
Corollaries
1.10,
2.8]
quoted
in
these
definitions.
Assertion
(ii)
follows,
for
instance,
by
comparing
the
given
global
T-pair
with
the
global
T-pair
data
M
T
(Π)
of
assertion
(iv)
via
the
reference
isomorphisms
that
appear
in
Definition
5.1,
(v).
In
light
of
assertion
(ii),
assertion
(iii)
is
immediate
from
the
definitions
[cf.
also
Proposition
3.2,
(ii),
(iv),
at
the
κ
nonarchimedean
v;
“κ
v
”,
the
Kummer
structure
of
“(X
v
M
v
)”
at
archimedean
v].
In
light
of
assertion
(iii),
assertion
(iv)
is
immediate
from
the
definitions
and
the
results
of
§1,
§2
[cf.,
especially,
Theorem
1.9;
Corollaries
1.10,
2.8]
quoted
in
these
definitions.
In
a
similar
vein,
assertions
(v),
(vi),
and
(vii)
are
immediate
from
the
definitions
and
the
results
quoted
in
these
definitions
[cf.
also
Proposition
3.2,
(ii),
κ
(iv),
at
the
nonarchimedean
v;
“κ
v
”,
the
Kummer
structure
of
“(X
v
M
v
)”
at
archimedean
v].
Remark
5.2.1.
Note
that
neither
of
the
composite
functors
EA
→
Th
,
EA
→
Th
T
of
Corollary
5.2,
(v),
(vi)
is
an
equivalence
of
categories!
Put
another
way,
there
is
no
natural,
functorial
way
to
“glue
together”
the
various
local
data
of
a
panalocal
Galois-theater/T-pair
so
as
so
obtain
a
“global
profinite
group”
that
determines
an
object
of
EA
.
∼
Remark
5.2.2.
By
applying
the
equivalence
EA
→
Th
of
Corollary
5.2,
(i),
one
may
obtain
a
factorization
EA
→
Th
→
An
[Th
T
]
of
the
functor
EA
→
An
[Th
T
]
of
Corollary
5.2,
(iv).
Thus,
we
obtain
equivalences
∼
∼
→
An
[Th
of
categories
Th
→
An
[Th
T
]
→
Th
;
the
functor
Th
T
]
may
be
thought
of
as
a
“global
analogue”
of
the
panalocal
functor
Th
→
An
[Th
T
]
of
Corollary
5.2,
(vii).
Remark
5.2.3.
A
similar
result
to
Corollary
5.2,
(ii)
[hence
also
similar
re-
sults
to
Corollary
5.2,
(iii),
(iv)],
may
be
obtained
when
T
=
TLG,
by
using
the
archimedean
primes,
which
are
“immune”
to
the
{±1}-indeterminacy
of
Proposi-
arc
tion
3.3,
(i).
Indeed,
in
the
notation
of
Definition
5.1,
(iii),
(v),
if
v
∈
V
,
then
by
combining
“κ
ell,v
ell
”
with
the
isomorphism
“ψ
v
”
arising
from
the
reference
iso-
morphism
of
the
global
Galois-theater
under
consideration
yields
an
inclusion
of
122
SHINICHI
MOCHIZUKI
∼
fields
k
NF
(Π)
→
A
X(Π,v
ell
)
→
A
X
v
.
On
the
other
hand,
by
applying
Corollary
1.10,
(c);
Remark
1.10.3,
(ii),
at
any
of
the
nonarchimedean
elements
of
V
,
it
follows
(Π)
may
be
related
to
the
roots
of
unity
of
k
NF
(Π),
while
the
restriction
that
μ
Z
morphism
at
v
of
the
global
T-pair
under
consideration,
together
with
the
Kummer
(M
)
to
the
roots
of
unity
of
A
X
v
.
Thus,
structure
at
v,
allow
one
to
relate
μ
Z
we
obtain
a
functorial
algorithm
[albeit
somewhat
more
complicated
than
the
al-
gorithm
discussed
in
Corollary
5.2,
(ii)]
for
constructing
the
natural
isomorphism
∼
(M
)
→
μ
(Π).
μ
Z
Z
Definition
5.3.
Let
T
∈
{TF,
TM,
TLG}.
We
shall
apply
a
subscript
“TLG”
(respectively,
“TCG”)
to
arithmetic
data
of
“T-pairs”
to
denote
the
result
of
ap-
plying
the
natural
functor
whose
codomain
is
the
corresponding
category
of
“TLG-
(respectively,
TCG)
pairs”
[cf.
Proposition
3.2;
Corollary
5.2].
In
the
following,
the
symbols
,
are
to
be
understood
as
shorthand
for
the
terms
“multiplicative”
and
“additive”,
respectively.
Let
κ
M
=
(V
,
M
,
{ρ
v
}
v∈V
,
{(Π
v
M
v
)}
v∈V
non
,
{(X
v
M
v
)}
v∈V
arc
)
def
be
a
global
T-pair
[where
V
is
as
in
Definition
5.1,
(iii)].
Thus,
M
is
equipped
with
a
natural
Aut(Π)-action
[cf.
Corollary
5.2,
(iv);
Remark
5.2.3].
In
the
follow-
ing,
we
shall
use
a
superscript
profinite
group
to
denote
the
sub-object
of
invariants
def
with
respect
to
that
profinite
group;
if
v
∈
V
=
V
/Aut(Π),
then
we
shall
write
M
v
for
the
arithmetic
data
of
the
[orbi-]MLF-Galois/Aut-holomorphic
T-pair
indexed
by
v
of
the
panalocal
T-pair
determined
by
M
,
and
Π
v
for
the
[orbi-]decomposition
group
of
v.
(i)
Suppose
that
T
=
TLG.
Then
a
-line
bundle
L
on
M
is
defined
to
be
a
collection
of
data
(L
[
];
{τ
[v]
∈
L
[v]
TV
}
v∈V
)
—
where
L
[
]
is
an
(M
)
Π
-torsor
equipped
with
an
Out(Π)-action
that
is
com-
patible
with
the
natural
Out(Π)-action
on
(M
)
Π
and,
moreover,
factors
through
the
quotient
Out(Π)
Im(Out(Π)
→
Out(Π/Δ));
for
each
v
∈
V
,
τ
[v]
∈
L
[v]
TV
def
is
a
trivialization
of
the
torsor
L
[v]
TV
over
(M
v
Π
v
)
TV
=
M
v
Π
v
/(M
v
Π
v
)
TCG
deter-
mined
by
the
M
v
Π
v
-torsor
L
[v]
obtained
from
L
[
]
via
ρ
v
,
for
v
∈
V
lying
over
v
—
such
that
any
element
of
L
[
]
determines
[by
restriction]
the
element
of
L
[v]
TV
given
by
τ
[v],
for
all
but
finitely
many
v
∈
V
.
[Here,
we
note
that
the
“[topological]
value
group”
(M
v
Π
v
)
TV
is
equipped
with
a
natural
ordering
[which
may
be
used
to
define
its
topology]
and
is
∼
=
R
if
v
∈
V
arc
;
moreover
the
nat-
=
Z
if
v
∈
V
non
and
∼
ural
ordering
on
(M
v
Π
v
)
TV
determines
a
natural
ordering
on
L
[v]
TV
.]
A
morphism
of
-line
bundles
on
M
ζ
:
L
1
→
L
2
TOPICS
IN
ABSOLUTE
ANABELIAN
GEOMETRY
III
123
∼
is
defined
to
be
an
Out(Π)-equivariant
isomorphism
ζ[
]
:
L
1
[
]
→
L
2
[
]
between
the
respective
(M
)
Π
-torsors
such
that
each
v
∈
V
induces
an
isomorphism
ζ[v]
TV
:
∼
L
1
[v]
TV
→
L
2
[v]
TV
that
maps
τ
1
[v]
to
an
element
of
L
2
[v]
TV
that
is
≤
τ
2
[v].
Write
Th
T
[M
]
for
the
category
of
-line
bundles
on
M
and
morphisms
of
-line
bundles
on
M
.
If
φ
:
M
1
→
M
2
is
a
morphism
of
global
T-pairs,
then
there
is
a
natural
pull-
back
functor
φ
∗
:
Th
T
[M
2
]
→
Th
T
[M
1
].
In
particular,
the
various
categories
Th
T
[M
]
together
form
a
fibered
category
Th
→
Th
T
T
over
Th
T
,
whose
fibers
are
the
categories
Th
T
[M
].
Finally,
we
observe
that
one
may
generalize
these
definitions
to
the
case
of
arbitrary
T
∈
{TF,
TM,
TLG}
by
applying
the
subscript
“TLG”,
where
necessary.
(ii)
Suppose
that
T
=
TF.
Write
O
M
for
the
ring
of
integers
of
the
field
M
.
Then
an
-line
bundle
L
on
M
is
defined
to
be
a
collection
of
data
(L
[
];
{|
−
|
L
[v]
}
v∈V
arc
)
Π
—
where
L
[
]
is
a
rank
one
projective
O
M
-module
equipped
with
an
Out(Π)-
action
that
is
compatible
with
the
natural
Out(Π)-action
on
(M
)
Π
and,
moreover,
factors
through
the
quotient
Out(Π)
Im(Out(Π)
→
Out(Π/Δ));
for
each
v
∈
V
arc
,
|
−
|
L
[v]
is
a
Hermitian
metric
on
the
M
v
-vector
space
L
[v]
obtained
from
L
[
]
⊗
(M
)
Π
via
ρ
v
,
for
v
∈
V
lying
over
v.
[Here,
we
recall
that
M
v
is
an
[orbi-]complex
archimedean
field.]
In
this
situation,
we
shall
also
write
L
[v]
for
the
M
v
Π
v
-vector
space
obtained
from
L
[
]
⊗
(M
)
Π
via
ρ
v
,
for
v
∈
V
lying
over
v
∈
V
non
.
In
Π
Π
particular,
the
O
M
-module
O
M
,
equipped
with
its
usual
Hermitian
metrics
at
arc
elements
of
V
,
determines
an
-line
bundle
which
we
shall
refer
to
as
the
trivial
-line
bundle.
A
morphism
of
-line
bundles
on
M
ζ
:
L
1
→
L
2
Π
is
defined
to
be
a
nonzero
Out(Π)-equivariant
morphism
of
O
M
]
:
-modules
ζ[
arc
L
1
[
]
→
L
2
[
]
such
that
for
each
v
∈
V
,
the
induced
isomorphism
ζ[v]
:
∼
L
1
[v]
→
L
2
[v]
maps
integral
elements
[i.e.,
elements
of
norm
≤
1]
with
respect
to
|
−
|
L
[v]
to
integral
elements
with
respect
to
|
−
|
L
[v]
.
Write
1
2
Th
T
[M
]
for
the
category
of
-line
bundles
on
M
and
morphisms
of
-line
bundles
on
M
.
If
φ
:
M
1
→
M
2
is
a
morphism
of
global
T-pairs,
then
there
is
a
natural
pull-
back
functor
φ
∗
:
Th
T
[M
2
]
→
Th
T
[M
1
].
In
particular,
the
various
categories
Th
T
[M
]
“glue
together”
to
form
a
fibered
category
Th
→
Th
T
T
124
SHINICHI
MOCHIZUKI
over
Th
T
,
whose
fibers
are
the
categories
Th
T
[M
].
Finally,
the
assignment
[in
the
notation
of
the
above
discussion]
L
[
]
→
the
(M
TLG
)
Π
-torsor
of
nonzero
sections
of
L
[
]
⊗
(M
)
Π
determines
[in
an
evident
fashion]
an
equivalence
of
categories
∼
Th
T
→
Th
T
over
Th
T
,
i.e.,
an
“equivalence
of
-
and
-line
bundles”.
(iii)
Let
∈
{,
};
if
=
,
then
assume
that
T
=
TF.
Then
observe
that
the
automorphism
group
of
any
object
of
Th
T
[M
]
is
naturally
isomorphic
)
Aut(Π)
.
To
avoid
various
problems
arising
to
the
finite
abelian
group
μ
Q/Z
(M
TLG
from
these
automorphisms,
it
is
often
useful
to
work
with
“coarsified
versions”
of
the
categories
introduced
in
(i),
(ii),
as
follows.
Write
|
|
Th
T
[M
]
for
the
[small,
id-rigid!]
category
whose
objects
are
isomorphism
classes
of
objects
Aut(Π)
of
Th
-orbits
of
morphisms
of
T
[M
]
and
whose
morphisms
are
μ
Q/Z
(M
TLG
)
Th
T
[M
].
Thus,
by
allowing
“M
”
to
vary,
we
obtain
a
fibered
category
|
|
Th
T
→
Th
T
|
|
over
Th
[M
].
Finally,
the
equivalence
of
T
,
whose
fibers
are
the
categories
Th
T
|
|
∼
||
categories
of
(ii)
determines
an
equivalence
of
categories
Th
T
→
Th
T
.
Remark
5.3.1.
In
the
notation
of
Definition
5.3,
(iii),
one
may
define
—
in
the
style
of
Corollary
5.2,
(iv)
—
a
category
An
[Th
T
,
||]
whose
objects
are
data
of
the
form
def
|
|
|
|
[M
M
T
(Π)
=
(M
T
(Π),
Th
T
T
[Π]])
for
Π
∈
Ob(EA
)
and
whose
morphisms
are
the
morphisms
induced
by
morphisms
|
|
of
EA
.
Here,
we
think
of
the
datum
“Th
T
[M
T
[Π]]”
as
an
object
of
the
category
whose
objects
are
small
categories
with
trivial
automorphism
groups
and
whose
morphisms
are
contravariant
functors.
Then,
just
as
in
Corollary
5.2,
(iv),
one
obtains
a
sequence
of
natural
functors
EA
→
An
[Th
T
,
||]
→
An
[Th
T
]
→
Th
T
→
EA
—
where
the
first
arrow
is
the
functor
obtained
by
assigning
Ob(EA
T
)
|
|
M
T
(Π)
—
all
of
which
are
equivalences
of
categories.
Definition
5.4.
Π
→
Let
T
∈
{TF,
TM},
•
∈
{
,
}.
(i)
If
Z
is
an
elliptically
admissible
hyperbolic
orbicurve
over
an
algebraic
clo-
sure
of
Q,
then
we
shall
refer
to
a
hyperbolic
orbicurve
X
as
in
Definition
5.1,
(ii),
TOPICS
IN
ABSOLUTE
ANABELIAN
GEOMETRY
III
125
as
geometrically
isomorphic
to
Z
if
[in
the
notation
of
loc.
cit.]
there
exists
an
isomorphism
of
schemes
X
F
∼
=
Z.
Write
EA
[Z]
⊆
EA
for
the
full
subcategory
determined
by
the
profinite
groups
isomorphic
to
Π
X
for
some
X
as
in
Definition
5.1,
(i),
that
is
geometrically
isomorphic
to
Z.
This
full
subcategory
determines,
in
an
evident
fashion,
full
subcategories
Th
•
[Z]
⊆
Th
•
;
Th
•
T
[Z]
⊆
Th
•
T
—
as
well
as
full
subcategories
of
the
“An
[−]”
versions
of
these
categories
discussed
in
Corollary
5.2
and
the
“-,
-line
bundle
versions”
discussed
in
Remark
5.3.1
[cf.
also
the
“measure-theoretic
versions”
discussed
in
Remark
5.9.1
below].
(ii)
By
applying
the
functors
“log
T,T
”
of
Proposition
3.2,
(v);
Proposition
4.2,
(ii),
to
the
various
local
data
of
a
panalocal
T-pair,
we
obtain
a
panalocal
log-
Frobenius
functor
log
T,T
:
Th
T
→
Th
T
which
is
naturally
isomorphic
to
the
identity
functor,
hence,
in
particular,
an
equiv-
alence
of
categories.
Note
that
the
construction
underlying
this
functor
leaves
the
underlying
panalocal
Galois-theater
unchanged,
i.e.,
log
T,T
“lies
over”
Th
.
Now
suppose
that
κ
M
=
(V
,
M
,
{ρ
v
}
v∈V
,
{(Π
v
M
v
)}
v∈V
non
,
{(X
v
M
v
)}
v∈V
arc
)
def
is
a
global
T-pair
[where
V
is
as
in
Definition
5.1,
(iii)].
Note
that
the
various
restriction
morphisms
ρ
v
determine
a
Π-equivariant
embedding
M
→
M
v
T
v∈V
of
M
into
a
certain
product
of
local
data.
Thus,
by
applying
the
functors
“log
T,T
”
of
Proposition
3.2,
(v);
Proposition
4.2,
(ii),
to
the
various
local
data
of
M
[i.e.,
more
precisely:
the
data,
other
than
the
{ρ
v
},
that
is
indexed
by
v
∈
V
],
we
obtain
a
“local
log-Frobenius
functor
log
v
T,T
”
on
the
portion
of
a
global
T-pair
constituted
by
this
local
data
which
is
naturally
isomorphic
to
the
identity
functor.
More-
over,
by
composing
this
natural
isomorphism
to
the
identity
functor
with
the
above
embedding
of
M
,
we
obtain
a
new
Π-equivariant
embedding
M
→
log
v
T,T
(M
v
T
)
v∈V
of
M
into
the
product
[as
above]
that
arises
from
the
output
“log
v
T,T
(M
v
T
)”
of
log
v
T,T
.
In
particular,
by
taking
the
image
of
this
new
embedding
to
be
the
global
data
∈
Ob(T
)
[i.e.,
the
“M
”]
of
a
new
global
T-pair
whose
local
data
is
given
by
applying
log
v
T,T
to
the
local
data
of
M
,
we
obtain
a
global
log-Frobenius
functor
log
T,T
:
Th
T
→
Th
T
126
SHINICHI
MOCHIZUKI
which
is
naturally
isomorphic
to
the
identity
functor,
hence,
in
particular,
an
equiv-
alence
of
categories.
Moreover,
the
construction
underlying
this
functor
leaves
the
underlying
global
Galois-theater
unchanged,
i.e.,
log
T,T
“lies
over”
Th
.
In
the
fol-
lowing
discussion,
we
shall
often
denote
[by
abuse
of
notation]
the
restriction
of
log
•
T,T
to
the
categories
“(−)[Z]”
by
log
•
T,T
.
Note
that
if
one
restricts
to
the
cat-
egories
“(−)[Z]”,
then
the
set
“V
/Aut(Π)”
has
a
meaning
which
is
independent
of
the
choice
of
a
particular
object
of
one
of
these
categories
[cf.
the
discussion
of
def
Definition
5.1,
(ii)].
In
the
following,
let
us
fix
a
v
∈
V
=
V
/Aut(Π).
(iii)
Consider,
in
the
notation
of
Definition
3.1,
(iv),
the
commutative
diagram
of
natural
maps
O
×
→
⏐
k
⏐
shell
post-log...
k
∼
id
−→
k
∼
→
×
k
⏐
⏐
→
k
...space-link
×
(k
)
pf
—
where
we
recall
that
k
∼
=
(O
×
)
pf
—
a
diagram
which
determines
an
oriented
k
graph
Γ
log
[i.e.,
whose
vertices
and
oriented
edges
correspond,
respectively,
to
the
non
objects
and
arrows
of
the
above
diagram];
write
Γ
non
(respectively,
Γ
non
)
for
the
oriented
subgraph
of
Γ
log
non
obtained
by
removing
the
upper
right-hand
arrow
“→
k”
id
(respectively,
the
lower
left-hand
arrow
“k
∼
−→”)
and
Γ
×
non
for
the
intersection
of
∼
Γ
log
non
,
Γ
non
.
Let
us
refer
to
the
lower
left-hand
vertex
of
Γ
non
[i.e.,
the
first
“k
”]
log
as
the
post-log
vertex
and
to
the
other
vertices
of
Γ
non
as
pre-log
vertices;
also
we
shall
refer
to
the
upper
right-hand
vertex
of
Γ
log
non
[i.e.,
“k”]
as
the
space-link
vertex.
Here,
we
wish
to
think
of
the
pre-log
copy
of
“k
∼
”
as
an
object
[i.e.,
“(O
×
)
pf
”]
def
k
×
formed
from
k
and
of
the
post-log
copy
of
“k
∼
”
as
the
“new
field”
—
i.e.,
the
new
copy
of
the
space-link
vertex
“k”
—
obtained
by
applying
the
log-Frobenius
functor.
Observe
that
the
entire
diagram
Γ
log
non
may
be
considered
as
a
diagram
in
the
category
TS,
whereas
the
diagram
Γ
non
may
be
considered
either
as
a
diagram
in
the
category
TS
or
as
a
diagram
in
the
category
TS
[i.e.,
relative
to
the
additive
topological
group
structure
of
the
field
k
∼
].
Write
p
k
for
the
residue
characteristic
def
def
of
k;
set
p
∗
k
=
p
k
if
p
k
is
odd
and
p
∗
k
=
p
2
k
if
p
k
=
2.
Then
since
[as
is
well-known]
∼
the
p
k
-adic
logarithm
determines
a
bijection
1
+
p
∗
k
·
O
k
→
p
∗
k
·
O
k
,
it
follows
that
O
k
Π
∼
k
⊆
I
=
(p
∗
k
)
−1
·
I
∗
def
⊆
(k
∼
)
Π
k
—
where
the
superscript
“Π
k
”
denotes
the
submodule
of
Galois-invariants,
and
we
write
I
∗
for
the
image
of
O
k
×
=
(O
×
)
Π
k
⊆
O
×
via
the
left-hand
vertical
arrow
k
k
of
the
above
diagram,
i.e.,
in
essence,
the
compact
submodule
constituted
by
the
pre-log-shell
discussed
in
Definition
3.1,
(iv).
We
shall
refer
to
I
as
the
log-shell
of
Γ
×
non
and
to
the
left-hand
vertical
arrow
of
the
above
diagram
as
the
shell-arrow.
In
fact,
if
k
is
absolutely
unramified
and
p
k
is
odd,
then
we
have
an
equality
O
k
Π
∼
k
=
I
[cf.
Remark
5.4.2
below].
MLF-sB
MLF-sB
(iv)
Next,
let
us
suppose
that
v
∈
V
non
;
recall
the
categories
C
TS
,
C
TS
MLF-sB
MLF-sB
of
Definition
3.1,
(iii).
Thus,
we
have
natural
functors
C
TS
→
C
TS
→
TG.
TOPICS
IN
ABSOLUTE
ANABELIAN
GEOMETRY
III
127
Let
us
write
MLF-sB
)
×
Orb(TG),v
Th
•
[Z];
N
v
=
Orb(C
TS
def
MLF-sB
N
v
=
Orb(C
TS
)
×
Orb(TG),v
Th
•
[Z]
def
—
where
the
“,
v”
in
the
fibered
product
is
to
be
understood
as
referring
to
the
natural
functor
Th
•
[Z]
→
Orb(TG)
given
by
the
assignment
“V
•
→
Π
v
”
[cf.
Defi-
nition
5.1,
(iv),
(b)].
Thus,
we
have
natural
functors
N
v
→
N
v
→
Th
•
[Z].
Next,
def
log
×
def
×
def
def
=
Γ
non
,
Γ
v
=
Γ
non
,
Γ
in
the
notation
of
(iii),
let
us
set
Γ
log
v
v
=
Γ
non
,
Γ
v
=
Γ
non
.
Then
for
each
vertex
ν
of
Γ
×
v
,
by
assigning
to
“k”
or
“O
”
[i.e.,
depending
on
k
the
choice
of
T
∈
{TF,
TM}]
the
object
at
the
vertex
ν
of
the
diagram
of
(iii),
we
MLF-sB
,
hence
by
considering
the
portion
of
obtain
a
natural
functor
C
T
MLF-sB
→
C
TS
the
panalocal
or
global
T-pair
under
consideration
that
is
indexed
by
v
or
v
∈
V
lying
over
v,
a
natural
functor
λ
v,ν
:
Th
•
T
[Z]
→
N
v
.
In
a
similar
vein,
if
ν
is
either
the
space-link
or
the
post-log
vertex
of
Γ
log
v
,
then
by
assigning
to
“k”
or
“O
k
”
the
underlying
additive
topological
group
of
the
field
“k”
[cf.
the
functorial
algorithms
of
Corollary
1.10,
as
applied
in
Proposition
3.2,
(iii)],
we
obtain
a
natural
functor
MLF-sB
,
hence
by
considering
the
portion
of
the
panalocal
or
global
C
T
MLF-sB
→
C
TS
T-pair
under
consideration
that
is
indexed
by
v
or
v
∈
V
lying
over
v,
a
natural
functor
λ
v,ν
:
Th
•
T
[Z]
→
N
v
.
Thus,
in
summary,
we
obtain
natural
functors
λ
v,ν
:
Th
•
T
[Z]
→
N
v
;
λ
v,ν
:
Th
•
T
[Z]
→
N
v
—
where
the
latter
functor
is
obtained
by
composing
the
former
functor
with
the
natural
functor
N
v
→
N
v
—
that
“lie
over”
Th
•
[Z],
for
each
vertex
ν
of
Γ
log
v
.
(v)
Consider,
in
the
notation
of
Definition
4.1,
(iv),
the
commutative
diagram
of
natural
maps
post-log...
k
∼
id
−→
k
∼
shell
−→
k
×
→
k
...space-link
—
a
diagram
which
determines
an
oriented
graph
Γ
log
arc
[i.e.,
whose
vertices
and
oriented
edges
correspond,
respectively,
to
the
objects
and
arrows
of
the
above
log
diagram];
write
Γ
arc
(respectively,
Γ
arc
)
for
the
oriented
subgraph
of
Γ
arc
obtained
id
by
removing
the
arrow
“→
k”
on
the
right
(respectively,
the
arrow
“k
∼
−→”
on
the
left)
and
Γ
×
arc
for
the
intersection
of
Γ
arc
,
Γ
arc
.
Let
us
refer
to
the
vertex
of
∼
log
Γ
log
arc
given
by
the
first
“k
”
as
the
post-log
vertex
and
to
the
other
vertices
of
Γ
arc
as
pre-log
vertices;
also
we
shall
refer
to
the
vertex
of
Γ
log
arc
given
by
“k”
as
the
space-link
vertex.
Here,
we
wish
to
think
of
the
pre-log
copy
of
“k
∼
”
as
an
object
formed
from
k
×
and
of
the
post-log
copy
of
“k
∼
”
as
the
“new
field”
—
i.e.,
the
new
copy
of
the
space-link
vertex
“k”
—
obtained
by
applying
the
log-Frobenius
functor.
Observe
that
the
entire
diagram
Γ
log
arc
may
be
considered
as
a
diagram
in
the
category
TH,
whereas
the
diagram
Γ
arc
may
be
considered
either
as
a
diagram
in
the
category
TH
or
as
a
diagram
in
TH
[i.e.,
relative
to
the
additive
topological
group
structure
of
the
field
k
∼
].
Note
that
it
follows
from
well-known
properties
of
the
[complex]
logarithm
that
O
k
∼
=
1
·I
π
⊆
I
=
O
k
×
∼
·
I
∗
def
⊆
k
∼
128
SHINICHI
MOCHIZUKI
—
where
we
we
write
I
∗
for
the
uniquely
determined
“line
segment”
[i.e.,
more
precisely:
closure
of
a
connected
pre-compact
open
subset
of
a
one-parameter
sub-
group]
of
k
∼
which
is
preserved
by
multiplication
by
±1
and
whose
endpoints
differ
by
a
generator
of
Ker(k
∼
k
×
).
Thus,
I
∗
maps
bijectively,
except
for
the
end-
points
of
the
line
segment,
to
the
pre-log-shell
discussed
in
Definition
4.1,
(iv).
We
∼
×
shall
refer
to
I
as
the
log-shell
of
Γ
×
arc
and
to
the
arrow
k
k
as
the
shell-arrow.
Also,
we
observe
that
I
may
be
constructed
as
the
closure
of
the
union
of
the
im-
ages
of
I
∗
via
the
finite
order
automorphisms
of
the
Aut-holomorphic
group
k
∼
;
in
particular,
the
formation
of
I
from
I
∗
depends
only
on
the
structure
of
k
∼
as
an
object
of
TH.
hol
hol
(vi)
Next,
let
us
suppose
that
v
∈
V
arc
;
recall
the
categories
C
TH
,
C
TH
of
hol
hol
Definition
4.1,
(iii).
Thus,
we
have
natural
functors
C
TH
→
C
TH
→
EA.
Let
us
write
hol
)
×
Orb(EA),v
Th
•
[Z];
N
v
=
Orb(C
TH
def
hol
N
v
=
Orb(C
TH
)
×
Orb(EA),v
Th
•
[Z]
def
—
where
the
“,
v”
in
the
fibered
product
is
to
be
understood
as
referring
to
the
nat-
ural
functor
Th
•
[Z]
→
Orb(EA)
given
by
the
assignment
“V
•
→
X
v
”
[cf.
Definition
5.1,
(iv),
(c)].
Thus,
we
have
natural
functors
N
v
→
N
v
→
Th
•
[Z].
Next,
in
the
def
log
×
def
×
def
def
=
Γ
arc
,
Γ
v
=
Γ
arc
,
Γ
v
=
Γ
arc
,
Γ
v
=
Γ
arc
.
Then
for
notation
of
(v),
let
us
set
Γ
log
v
×
each
vertex
ν
of
Γ
v
,
by
assigning
to
“k”
or
“O
k
”
[i.e.,
depending
on
the
choice
of
T
∈
{TF,
TM}]
the
object
at
the
vertex
ν
of
the
diagram
of
(v),
we
obtain
a
natural
hol
,
hence
by
considering
the
portion
of
the
panalocal
or
global
functor
C
T
hol
→
C
TH
or
T-pair
under
consideration
that
is
indexed
by
v
or
v
∈
V
lying
over
v,
a
natural
functor
λ
v,ν
:
Th
•
T
[Z]
→
N
v
.
In
a
similar
vein,
if
ν
is
either
the
space-link
or
the
post-log
vertex
of
Γ
log
v
,
then
by
assigning
to
“k”
or
“O
k
”
the
underlying
additive
topological
group
of
the
field
“k”
[cf.
the
functorial
algorithms
of
Corollary
2.7,
as
hol
,
hence
by
applied
in
Proposition
4.2,
(ii)],
we
obtain
a
natural
functor
C
T
hol
→
C
TH
considering
the
portion
of
the
panalocal
or
global
T-pair
under
consideration
that
is
indexed
by
v
or
v
∈
V
lying
over
v,
a
natural
functor
λ
v,ν
:
Th
•
T
[Z]
→
N
v
.
Thus,
in
summary,
we
obtain
natural
functors
λ
v,ν
:
Th
•
T
[Z]
→
N
v
;
λ
v,ν
:
Th
•
T
[Z]
→
N
v
—
where
the
latter
functor
is
obtained
by
composing
the
former
functor
with
the
natural
functor
N
v
→
N
v
—
that
“lie
over”
Th
•
[Z],
for
each
vertex
ν
of
Γ
log
v
.
(vii)
Finally,
in
the
notation
of
(iv)
(respectively,
(vi))
for
v
∈
V
non
(respec-
log
tively,
v
∈
V
arc
):
For
each
edge
of
Γ
v
(respectively,
Γ
v
)
running
from
a
vertex
ν
1
to
a
vertex
ν
2
,
the
arrow
in
the
diagram
of
(iii)
(respectively,
(v))
corresponding
to
determines
a
natural
transformation
ι
v,
:
λ
v,ν
1
◦
Λ
ν
1
→
λ
v,ν
2
(respectively,
ι
v,
:
λ
v,ν
1
◦
Λ
ν
1
→
λ
v,ν
2
)
—
where,
for
each
pre-log
vertex
ν
of
Γ
log
v
,
we
take
Λ
ν
to
be
the
identity
functor
on
•
log
Th
T
[Z];
for
the
post-log
vertex
ν
of
Γ
v
,
we
take
Λ
ν
to
be
the
log-Frobenius
functor
log
•
T,T
:
Th
•
T
[Z]
→
Th
•
T
[Z].
TOPICS
IN
ABSOLUTE
ANABELIAN
GEOMETRY
III
129
Remark
5.4.1.
Note
that
the
diagrams
of
Definition
5.4,
(iii),
(v),
[hence
also
the
natural
transformations
of
Definition
5.4,
(vii)]
cannot
be
extended
to
global
number
fields!
Indeed,
this
observation
is,
in
essence,
a
reflection
of
the
fact
that
the
various
logarithms
that
may
be
defined
at
the
various
completions
of
a
number
field
do
not
induce
maps
from,
say,
the
group
of
units
of
the
number
field
to
the
number
field!
Remark
5.4.2.
Note
that
in
the
context
of
Definition
5.4,
(iii),
when
k
is
not
absolutely
unramified,
the
“gap”
between
O
k
Π
∼
k
and
I
may
be
bounded
in
terms
of
the
ramification
index
of
k
over
Q
p
k
.
We
leave
the
routine
details
to
the
interested
reader.
Remark
5.4.3.
The
inclusions
“O
k
Π
∼
k
⊆
I”,
“O
k
∼
⊆
I”
of
Definition
5.4,
(iii),
(v),
may
be
thought
of
as
inclusions,
within
the
log-shell
I,
of
the
various
localizations
of
the
trivial
-line
bundle
of
Definition
5.3,
(ii)
—
an
-line
bundle
whose
structure
is
determined
by
the
global
ring
of
integers
[i.e.,
“O
M
”
in
the
notation
of
Definition
5.3,
(ii)],
equipped
its
natural
metrics
at
the
archimedean
primes.
That
is
to
say,
the
definition
of
the
trivial
-line
bundle
involves,
in
an
essential
way,
not
just
the
additive
[i.e.,
“”]
structure
of
the
global
ring
of
integers,
but
also
the
multiplicative
[i.e.,
“”]
structure
of
the
global
ring
of
integers.
Next,
we
consider
the
following
global/panalocal
analogue
of
Corollaries
3.6,
4.5.
Corollary
5.5.
(Global
and
Panalocal
Mono-anabelian
Log-Frobenius
Compatibility)
Let
Z
be
an
elliptically
admissible
hyperbolic
orbicurve
over
an
algebraic
closure
of
Q,
with
field
of
moduli
F
mod
[cf.
Definition
5.1,
(ii)];
T
∈
{TF,
TM};
•
∈
{
,
}.
Set
X
=
Th
•
T
[Z],
E
•
=
Th
•
[Z].
Consider
the
diagram
of
categories
D
•
def
...
...
...
log
X
−→
id
+1
λ
v
...
X
⏐
⏐
id
X
⏐
⏐
⏐
⏐
.
.
.
λ
v
def
log
−→
X
...
id
−1
...
.
.
.
λ
v
...
N
v
⏐
⏐
N
v
⏐
⏐
...
...
N
v
⏐
⏐
...
N
v
N
v
...
...
N
v
⏐
⏐
...
...
E
•
⏐
⏐
κ
•
An
An
•
[X
]
...
130
SHINICHI
MOCHIZUKI
⏐
⏐
E
•
—
where
we
use
the
notation
“log”
for
the
evident
restriction
of
the
arrows
“log
•
T,T
”
•
of
Definition
5.4,
(ii);
for
positive
integers
n
≤
7,
we
shall
denote
by
D
≤n
the
•
subdiagram
of
categories
of
D
determined
by
the
first
n
[of
the
seven]
rows
of
D
•
;
we
write
L
for
the
countably
ordered
set
determined
[cf.
§0]
by
the
infinite
linear
oriented
graph
Γ
opp
D
•
[so
the
elements
of
L
correspond
to
vertices
of
the
≤1
first
row
of
D
•
]
and
L
†
=
L
∪
{}
def
for
the
ordered
set
obtained
by
appending
to
L
a
formal
symbol
[which
we
think
of
as
corresponding
to
the
unique
vertex
of
the
second
row
of
D
•
]
such
that
<
,
for
all
∈
L;
id
denotes
the
identity
functor
at
the
vertex
∈
L;
the
vertices
of
the
third
and
fourth
rows
of
D
•
are
indexed
by
the
elements
v
,
v,
v
,
.
.
.
of
the
set
of
valuations
V(F
mod
)
of
F
mod
;
the
arrows
from
the
second
row
to
the
def
category
N
v
in
the
third
row
are
given
by
the
collection
of
functors
λ
v
=
{λ
v,ν
}
ν
of
Definition
5.4,
(iv),
(vi),
where
ν
ranges
over
the
pre-log
vertices
of
Γ
log
[or,
v
log
alternatively,
over
all
the
vertices
of
Γ
v
,
subject
to
the
proviso
that
we
identify
the
functors
associated
to
the
space-link
and
post-log
vertices];
the
arrows
from
the
third
to
fourth
and
from
the
fourth
to
fifth
rows
are
the
natural
functors
N
v
→
N
v
→
E
•
of
Definition
5.4,
(iv),
(vi);
the
arrows
from
the
fifth
to
sixth
and
from
the
sixth
to
seventh
rows
are
the
natural
equivalences
of
categories
∼
∼
E
•
→
An
•
[X
]
→
E
•
,
the
first
of
which
we
shall
denote
by
κ
An
•
,
of
Corollary
5.2,
(i),
(iv),
(vii)
[cf.
also
Remark
5.2.2],
restricted
to
“[Z]”;
we
shall
apply
“[−]”
to
the
names
of
arrows
appearing
in
D
•
to
denote
the
path
of
length
1
associated
to
the
arrow.
Also,
let
us
write
∼
φ
An
•
:
An
•
[X
]
→
X
for
the
equivalence
of
categories
given
by
the
“forgetful
functor”
of
Corollary
5.2,
(iv),
(vii),
restricted
to
“[Z]”,
π
An
•
:
X
→
An
•
[X
]
for
the
quasi-inverse
for
φ
An
•
given
by
the
composite
of
the
natural
projection
functor
X
→
E
•
with
κ
An
•
:
∼
E
•
→
An
•
[X
],
and
η
An
•
:
φ
An
•
◦
π
An
•
→
id
X
for
the
isomorphism
that
exhibits
φ
An
•
,
π
An
•
as
quasi-inverses
to
one
another.
Then:
•
•
admits
a
natural
structure
of
core
on
D
≤n−1
.
That
is
(i)
For
n
=
5,
6,
7,
D
≤n
•
•
to
say,
loosely
speaking,
E
,
An
[X
]
“form
cores”
of
the
functors
in
D.
If,
moreover,
•
=
,
then
one
obtains
a
natural
structure
of
core
on
D
•
by
appending
to
the
final
row
of
D
•
the
natural
arrow
E
•
→
EA
[Z].
(ii)
The
“forgetful
functor”
φ
An
•
gives
rise
to
a
telecore
structure
T
An
•
•
on
D
≤5
,
whose
underlying
diagram
of
categories
we
denote
by
D
An
•
,
by
appending
•
to
D
≤6
telecore
edges
...
φ
+1
...
X
An
•
[X
]
⏐
⏐
φ
log
−→
X
log
−→
φ
−1
...
X
...
TOPICS
IN
ABSOLUTE
ANABELIAN
GEOMETRY
III
An
•
[X
]
φ
−→
131
X
•
from
the
core
An
•
[X
]
to
the
various
copies
of
X
in
D
≤2
given
by
copies
of
φ
An
•
,
†
which
we
denote
by
φ
,
for
∈
L
.
That
is
to
say,
loosely
speaking,
φ
An
•
de-
•
0
.
Finally,
for
each
∈
L
†
,
let
us
write
[β
]
termines
a
telecore
structure
on
D
≤5
1
for
the
path
on
Γ
D
An
•
of
length
0
at
and
[β
]
for
some
[cf.
the
coricity
of
(i)!]
path
on
Γ
D
An
•
of
length
∈
{5,
6}
[i.e.,
depending
on
whether
or
not
=
]
that
starts
from
,
descends
via
some
path
of
length
∈
{4,
5}
to
the
core
vertex
“An
•
[X
]”,
and
returns
to
via
the
telecore
edge
φ
.
Then
the
collection
of
natural
transformations
−1
−1
,
η
,
η
}
∈L,∈L
†
{η
,
η
—
where
we
write
η
for
the
identity
natural
transformation
from
the
arrow
φ
:
An
•
[X
]
→
X
to
the
composite
arrow
id
◦
φ
:
An
•
[X
]
→
X
and
∼
η
:
(D
An
•
)
[β
1
]
→
(D
An
•
)
[β
0
]
for
the
isomorphism
arising
from
η
An
•
—
generate
a
contact
structure
H
An
•
on
the
telecore
T
An
•
.
(iii)
The
natural
transformations
[cf.
Definition
5.4,
(vii)]
ι
v,
:
λ
v,ν
1
◦
Λ
ν
1
→
λ
v,ν
2
(respectively,
ι
v,
:
λ
v,ν
1
◦
Λ
ν
1
→
λ
v,ν
2
)
log
—
where
v
∈
V(F
mod
);
is
an
edge
of
Γ
v
(respectively,
Γ
v
)
running
from
a
vertex
ν
1
to
a
vertex
ν
2
;
if
ν
1
is
a
pre-log
vertex,
then
we
interpret
the
domain
and
codomain
of
ι
v,
(respectively,
ι
v,
)
as
the
arrows
associated
to
the
paths
of
length
1
(respectively,
2)
from
the
second
to
third
(respectively,
fourth)
rows
of
D
•
determined
by
v
and
ν
1
,
ν
2
;
if
ν
1
is
a
post-log
vertex,
then
we
interpret
the
domain
of
ι
v,
(respectively,
ι
v,
)
as
the
arrow
associated
to
the
path
of
length
3
(respectively,
4)
from
the
first
to
the
third
(respectively,
fourth)
rows
of
D
•
determined
by
v,
ν
1
,
and
the
condition
that
the
initial
length
2
portion
of
the
path
be
a
path
of
the
form
[id
]
◦
[log]
[for
∈
L],
and
we
interpret
the
codomain
of
ι
v,
(respectively,
ι
v,
)
as
the
arrow
associated
to
the
path
of
length
2
(respectively,
3)
from
the
first
to
the
third
(respectively,
fourth)
rows
of
D
•
determined
by
v,
ν
2
,
and
the
condition
that
the
initial
length
1
portion
of
the
path
be
a
path
of
the
form
[id
+1
]
[for
the
•
•
(respectively,
D
≤4
)
same
∈
L]
—
belong
to
a
family
of
homotopies
on
D
≤3
•
•
that
determines
on
the
portion
of
D
≤3
(respectively,
D
≤4
)
indexed
by
v
a
structure
•
•
of
observable
S
log
(respectively,
S
log
)
on
D
≤2
(respectively,
the
portion
of
D
≤3
indexed
by
v).
Moreover,
the
families
of
homotopies
that
constitute
S
log
and
S
log
are
compatible
with
one
another
as
well
as
with
the
families
of
homotopies
that
constitute
the
core
and
telecore
structures
of
(i),
(ii).
•
•
does
not
admit
a
structure
of
core
on
D
≤1
(iv)
The
diagram
of
categories
D
≤2
which
[i.e.,
whose
constituent
family
of
homotopies]
is
compatible
with
[the
con-
stituent
family
of
homotopies
of
]
the
observables
S
log
,
S
log
of
(iii).
Moreover,
the
telecore
structure
T
An
•
of
(ii),
the
contact
structure
H
An
•
of
(ii),
and
the
observables
S
log
,
S
log
of
(iii)
are
not
simultaneously
compatible.
(v)
The
unique
vertex
of
the
second
row
of
D
•
is
a
nexus
of
Γ
•D
.
More-
over,
D
•
is
totally
-rigid,
and
the
natural
action
of
Z
on
the
infinite
linear
132
SHINICHI
MOCHIZUKI
•
oriented
graph
Γ
D
≤1
extends
to
an
action
of
Z
on
D
•
by
nexus-classes
of
self-
equivalences
of
D
•
.
Finally,
the
self-equivalences
in
these
nexus-classes
are
com-
patible
with
the
families
of
homotopies
that
constitute
the
cores
and
observ-
ables
of
(i),
(iii);
these
self-equivalences
also
extend
naturally
[cf.
the
technique
of
extension
applied
in
Definition
3.5,
(vi)]
to
the
diagram
of
categories
[cf.
Definition
3.5,
(iv),
(a)]
that
constitutes
the
telecore
of
(ii),
in
a
fashion
that
is
compatible
with
both
the
family
of
homotopies
that
constitutes
this
telecore
structure
[cf.
Definition
3.5,
(iv),
(b)]
and
the
contact
structure
H
An
•
of
(ii).
(vi)
There
is
a
natural
panalocalization
morphism
of
diagrams
of
categories
D
→
D
[cf.
the
panalocalization
functors
of
Definition
5.1,
(iv),
(vi)]
that
lies
over
the
∼
evident
isomorphism
of
oriented
graphs
Γ
D
→
Γ
D
and
is
compatible
with
the
cores
of
(i),
the
telecore
and
contact
structures
of
(ii),
the
observables
of
(iii),
and
the
Z-actions
of
(v).
Proof.
Assertions
(i),
(ii)
are
immediate
from
the
definitions
—
cf.
also
the
proofs
of
Corollary
3.6,
(i),
(ii);
Corollary
4.5,
(i),
(ii).
Next,
we
consider
assertion
(iii).
def
The
data
arising
from
applying
the
collection
of
functors
λ
v
=
{λ
v,ν
}
ν
to
the
data
arising
from
id
,
as
ranges
over
the
elements
of
L,
yields
a
diagram
of
copies
of
Γ
log
indexed
by
elements
of
L
v
...
(
Γ
log
v
)
+1
(
Γ
log
v
)
(
Γ
log
v
)
−1
...
log
—
where
the
symbol
“”
denotes
the
result
of
gluing
(
Γ
log
v
)
+1
onto
(
Γ
v
)
by
log
identifying
the
post-log
vertex
of
(
Γ
log
v
)
+1
with
the
space-link
vertex
of
(
Γ
v
)
.
Now
the
existence
of
a
family
of
homotopies
as
asserted
follows,
in
a
routine
fashion,
from
the
fact
that
the
above
diagram
is
commutative
[i.e.,
one
does
not
obtain
any
pairs
of
distinct
maps
by
traveling
along
distinct
co-verticial
pairs
of
paths
of
the
diagram]
—
cf.
the
relationship
of
the
diagrams
×
...
←
k
+1
...
←
×
k
+1
→
×
(k
+1
)
pf
∼
k
+1
←
×
k
×
←
k
→
×
(k
)
pf
←
.
.
.
∼
k
←
.
.
.
of
Remarks
3.6.1,
(i);
4.5.1,
(i),
to
the
proofs
of
assertion
(iii)
of
Corollaries
3.6,
4.5;
we
leave
the
routine
details
[which
are
entirely
similar
to
the
proofs
of
assertion
(iii)
of
Corollaries
3.6,
4.5]
to
the
reader.
Finally,
the
compatibility
of
the
resulting
family
of
homotopies
with
the
families
of
homotopies
that
constitute
the
core
and
telecore
structures
of
(i),
(ii)
is
immediate
from
the
definitions.
This
completes
the
proof
of
assertion
(iii).
Next,
we
consider
assertion
(iv).
Recall
that
the
proofs
of
the
incompatibil-
ity
assertions
of
assertion
(iv)
of
Corollaries
3.6,
4.5
amount,
in
essence,
to
the
incompatibility
of
the
introduction
of
a
single
model
that
maps
isomorphically
to
various
copies
of
the
model
indexed
by
elements
of
L
in
the
diagrams
of
Remarks
3.6.1,
(i);
4.5.1,
(i).
In
the
present
situation,
the
incompatibility
assertions
of
as-
sertion
(iv)
of
the
present
Corollaries
5.5
amount,
in
an
entirely
similar
fashion,
to
TOPICS
IN
ABSOLUTE
ANABELIAN
GEOMETRY
III
133
log
that
maps
the
incompatibility
of
the
introduction
of
a
single
model
(
Γ
log
v
)
of
Γ
v
isomorphically
(
Γ
log
v
)
...
...
(
Γ
log
v
)
+1
↓
(
Γ
log
v
)
(
Γ
log
v
)
−1
...
...
to
each
copy
(
Γ
log
v
)
that
appears
in
the
diagram
that
was
used
in
the
proof
of
assertion
(iii).
We
leave
the
routine
details
[which
are
entirely
similar
to
the
proofs
of
assertion
(iv)
of
Corollaries
3.6,
4.5]
to
the
reader.
This
completes
the
proof
of
assertion
(iv).
Next,
we
consider
assertion
(v).
The
fact
that
is
a
nexus
of
Γ
•D
is
immediate
from
the
definitions.
When
•
=
,
the
total
-rigidity
of
D
•
follows
immediately
∼
from
the
equivalence
of
categories
Th
T
→
EA
of
Corollary
5.2,
(iv),
together
with
the
slimness
of
the
profinite
groups
that
appear
as
objects
of
EA
[cf.,
e.g.,
[Mzk20],
Proposition
2.3,
(ii)].
When
•
=
,
the
total
-rigidity
of
D
•
follows,
in
light
of
the
“Kummer
theory”
of
Propositions
3.2,
(iv);
4.2,
(i),
from
the
fact
that
the
orbi-objects
that
appear
in
the
definition
of
a
panalocal
Galois-theater
are
defined
in
such
a
way
as
to
eliminate
all
the
automorphisms
[cf.
Definition
5.1,
(iv),
(b),
(c)].
The
remainder
of
assertion
(v)
is
immediate
from
the
definitions
and
constructions
made
thus
far.
This
completes
the
proof
of
assertion
(v).
Finally,
we
observe
that
assertion
(vi)
is
immediate
from
the
definitions
and
constructions
made
thus
far.
Remark
5.5.1.
The
“general
formal
content”
of
the
remarks
following
Corollaries
3.6,
3.7
applies
to
the
situation
discussed
in
Corollary
5.5,
as
well.
We
leave
the
routine
details
of
translating
these
remarks
into
the
language
of
the
situation
of
Corollary
5.5
to
the
interested
reader.
Remark
5.5.2.
Note
that
it
does
not
appear
realistic
to
attempt
to
construct
a
theory
of
“geometric
panalocalization”
with
respect
to
the
various
closed
points
of
the
hyperbolic
orbicurve
over
an
MLF
under
consideration
[cf.
the
discussion
of
Remarks
1.11.5;
3.7.7,
(ii)].
Indeed,
the
decomposition
groups
of
such
closed
points
[which
are
either
isomorphic
to
the
absolute
Galois
group
of
an
MLF
or
an
extension
of
such
an
absolute
Galois
group
by
a
copy
of
Z(1)]
do
not
satisfy
an
appropriate
analogue
of
the
crucial
mono-anabelian
result
Corollary
1.10
[hence,
in
particular,
do
not
lead
to
a
situation
in
which
both
of
the
two
combinatorial
dimensions
of
the
absolute
Galois
group
of
an
MLF
under
consideration
are
rigidified
—
cf.
Remark
1.9.4].
Definition
5.6.
(i)
Recall
the
categories
TG,
TM,
and
TS
of
Definition
3.1,
(i),
(iii).
Write
TG
⊆
TG
134
SHINICHI
MOCHIZUKI
for
the
subcategory
given
by
the
profinite
groups
isomorphic
to
the
absolute
Galois
group
of
an
MLF
and
open
injections
of
profinite
groups,
and
TG
cs
⊆
Orb(TG)
for
the
full
subcategory
determined
by
the
[“coarsified”]
objects
of
Orb(TG)
obtained
by
considering
an
object
G
∈
Ob(TG)
up
to
its
group
of
automorphisms
Aut
TG
(G).
Write
TM
−
→
for
the
category
whose
objects
(C,
C
)
consist
of
a
topological
monoid
C
isomorphic
−
→
to
O
C
and
a
topological
submonoid
C
⊆
C
[necessarily
isomorphic
to
R
≥0
]
such
that
the
natural
inclusions
C
×
→
C
[where
C
×
,
which
is
necessarily
isomorphic
to
−
→
S
1
,
denotes
the
topological
submonoid
of
invertible
elements],
C
→
C
determine
an
isomorphism
−
→
∼
C
×
×
C
→
C
−
→
−
→
of
topological
monoids,
and
whose
morphisms
(C
1
,
C
1
)
→
(C
2
,
C
2
)
are
isomor-
−
→
∼
−
→
∼
phisms
of
topological
monoids
C
1
→
C
2
that
induce
isomorphisms
C
1
→
C
2
.
If
G
∈
Ob(TG),
then
let
us
write
Lie(G)
for
the
associated
group
germ
—
i.e.,
the
associated
group
pro-object
of
TS
de-
termined
by
the
neighborhoods
of
the
identity
element
—
and,
when
G
is
abelian,
Lie
±
(G)
for
the
orbi-group
germ
obtained
by
working
with
Lie(G)
up
to
“{±1}”.
Write
TB
for
the
category
whose
objects
(B,
B
,
B
,
β)
consist
of
a
two-dimensional
connected
topological
Lie
group
B
equipped
with
two
one-parameter
subgroups
B
,
B
⊆
B
that
determine
an
isomorphism
∼
B
×
B
→
B
∼
of
topological
groups,
together
with
an
isomorphism
β
:
Lie
±
(B
)
→
Lie
±
(B
),
and
whose
morphisms
(B
1
,
B
1
,
B
1
,
β
1
)
→
(B
2
,
B
2
,
B
2
,
β
2
)
are
the
surjective
homomor-
phisms
B
1
→
B
2
of
topological
groups
that
are
compatible
with
the
B
i
,
B
i
,
β
i
for
i
=
1,
2.
Write
TB
for
the
category
of
orientable
topological
orbisurfaces
[i.e.,
which
are
topological
surfaces
over
the
complement,
in
the
“coarse
space”
associated
to
the
orbisurface,
of
some
discrete
closed
subset]
and
local
isomorphisms
between
such
orbisurfaces.
Thus,
we
obtain
natural
“forgetful
functors”
TM
→
TM;
TB
→
TB
−
→
determined
by
the
assignments
(C,
C
)
→
C,
(B,
B
,
B
,
β)
→
B,
as
well
as
natural
“decomposition
functors”
dec
TM
:
TM
→
TG
×
TG
cs
;
dec
TB
:
TB
→
TG
×
TG
cs
−
→
−
→
gp
determined
by
the
assignments
(C,
C
)
→
(C
×
,
(
C
)
cs
),
(B,
B
,
B
,
β)
→
(B
,
(B
)
cs
)
[where
“gp”
denotes
the
groupification
of
a
monoid;
“cs”
denotes
the
result
of
con-
sidering
a
topological
group
up
to
its
group
of
automorphisms].
TOPICS
IN
ABSOLUTE
ANABELIAN
GEOMETRY
III
135
(ii)
We
shall
refer
to
as
a
mono-analytic
Galois-theater
any
collection
of
data
W
=
(W
,
{(G
w
,
|Π|
w
)}
w∈W
non
,
{(G
w
,
|X|
w
)}
w∈W
arc
)
def
—
where
W
is
a
set
that
admits
a
decomposition
as
a
disjoint
union
W
=
def
{
W
}
W
arc
⊇
W
=
W
non
W
arc
;
for
each
w
∈
W
non
,
G
w
∈
W
non
Ob(Orb(TG
)),
and
|Π|
w
is
an
isomorphism
class
of
pro-objects
of
the
category
TG;
for
each
w
∈
W
arc
,
G
w
∈
Ob(Orb(TM
)),
and
|X|
w
is
an
isomorphism
class
of
EA
—
such
that
there
exists
a
panalocal
Galois-theater
V
=
(V
,
{Π
v
}
v∈V
non
,
{X
v
}
v∈V
arc
)
def
[cf.
the
notation
of
Definition
5.1,
(iv)]
and
an
isomorphism
of
sets
∼
ψ
W
:
V
→
W
—
which
we
shall
refer
to
as
a
reference
isomorphism
for
W
—
that
satisfies
the
∼
∼
following
conditions:
(a)
ψ
W
maps
V
→
W
,
V
non
→
W
non
,
V
arc
→
W
arc
;
(b)
for
each
v
∈
V
non
mapping
to
w
∈
W
non
,
G
w
is
isomorphic
to
the
[group-
theoretically
characterizable
—
cf.
Remark
1.9.2]
quotient
Π
v
G
v
determined
by
the
absolute
Galois
group
of
the
base
field,
and
the
class
|Π|
w
contains
the
pro-object
of
TG
determined
by
the
projective
system
of
open
subgroups
of
Π
v
arising
from
open
subgroups
of
G
v
;
(c)
for
each
v
∈
V
arc
mapping
to
w
∈
W
arc
,
X
v
belongs
to
the
class
|X|
w
,
and
G
w
is
isomorphic
to
the
object
of
TM
determined
R
>0
).
A
morphism
of
mono-analytic
Galois-theaters
,
O
A
by
(O
A
X
v
X
v
φ
:
(W
1
,{((G
1
)
w
1
,
|Π|
w
1
)}
w
1
∈W
1
non
,
{((G
1
)
w
1
,
|X|
w
1
)}
w
1
∈W
1
arc
)
→
(W
2
,
{((G
2
)
w
2
,
|Π|
w
2
)}
w
2
∈W
2
non
,
{((G
2
)
w
2
,
|X|
w
2
)}
w
2
∈W
2
arc
)
∼
is
defined
to
consist
of
a
bijection
of
sets
φ
W
:
W
1
→
W
2
that
induces
bijec-
∼
∼
tions
W
1
non
→
W
2
non
,
W
1
arc
→
W
2
arc
that
are
compatible
with
the
isomorphism
classes
|Π|
w
i
,
|X|
w
i
[for
i
=
1,
2],
together
with
open
injections
of
[orbi-]profinite
w
1
→
w
2
∈
W
2
non
],
and
isomorphisms
groups
(G
1
)
w
1
→
(G
2
)
w
2
[where
W
1
non
∼
(G
1
)
w
1
→
(G
2
)
w
2
[where
W
1
arc
w
1
→
w
2
∈
W
2
arc
].
Write
Th
for
the
category
of
mono-analytic
Galois-theaters
and
morphisms
of
mono-analytic
Galois-theaters.
Thus,
if
Z
is
an
elliptically
admissible
hyperbolic
orbicurve
over
an
algebraic
clo-
sure
of
Q,
then
we
have
a
full
subcategory
Th
[Z]
⊆
Th
,
together
with
natural
“mono-analyticization
functors”
Th
→
Th
;
Th
[Z]
→
Th
[Z]
—
which
are
essentially
surjective.
(iii)
Next,
let
us
fix
a
mono-analytic
Galois-theater
W
as
in
(ii),
together
with
MLF
MLF
,
C
TS
of
Definition
3.1,
(iii).
Thus,
we
a
w
∈
W
non
.
Recall
the
categories
C
TS
have
a
[1-]commutative
diagram
of
natural
functors
MLF-sB
C
TS
⏐
⏐
MLF-sB
−→
C
TS
⏐
⏐
−→
TG
sB
⏐
⏐
MLF
C
TS
−→
−→
MLF
C
TS
TG
136
SHINICHI
MOCHIZUKI
—
in
which
the
vertical
arrows
are
“mono-analyticization
functors”
[cf.
the
mono-
analyticization
functors
of
(ii);
the
construction
implicit
in
(ii),
(b)];
the
arrows
MLF-sB
MLF
→
TG
sB
,
C
TS
→
TG
are
the
natural
projection
functors.
Let
us
write
C
TS
def
MLF
=
Orb(C
TS
)
×
Orb(TG
),w
Th
[Z]
N
w
def
MLF
N
w
=
Orb(C
TS
)
×
Orb(TG
),w
Th
[Z]
—
where
the
“,
w”
in
the
fibered
product
is
to
be
understood
as
referring
to
the
natural
functor
Th
[Z]
→
Orb(TG
)
given
by
the
assignment
“W
→
G
w
”
[cf.
(ii)].
Thus,
we
have
natural
functors
N
w
→
N
w
→
Th
[Z].
(iv)
Next,
let
us
fix
a
mono-analytic
Galois-theater
W
as
in
(ii),
together
with
hol
hol
,
C
TH
of
Definition
4.1,
(iii).
Write
a
w
∈
W
arc
.
Recall
the
categories
C
TH
hol
C
TB
for
the
category
whose
objects
are
triples
(G,
M,
κ
M
),
where
G
∈
Ob(TM
),
M
∈
Ob(TB),
and
κ
M
:
dec
TB
(M
)
dec
TM
(G)
—
which
we
shall
refer
to
as
the
Kummer
structure
of
the
object
—
is
a
pair
of
surjective
homomorphisms
of
TG,
TG
cs
,
and
whose
morphisms
φ
:
(G
1
,
M
1
,
κ
M
1
)
→
(G
2
,
M
2
,
κ
M
2
)
consist
of
an
∼
isomorphism
φ
G
:
G
1
→
G
2
of
TM
and
a
morphism
φ
M
:
M
1
→
M
2
of
TB
that
hol
def
are
compatible
with
κ
M
1
,
κ
M
2
;
write
C
TB
=
TM
×
TB.
Next:
κ
hol
)
[cf.
Definition
4.1,
(i)].
Recall
Suppose
that
(X
ell
M
k
)
∈
Ob(C
TH
κ
that
the
Kummer
structure
of
(X
ell
M
k
)
consists
of
an
Aut-holomorphic
homomorphism
from
M
k
to
an
isomorph
of
“C
×
(
∼
=
S
1
×
R
>0
)”;
observe
that
the
Aut-holomorphic
automorphisms
of
Lie(C
×
)
of
order
4
determine
∼
an
isomorphism
Lie
±
(S
1
×{1})
→
Lie
±
({1}×R
>0
).
Thus,
by
pulling
back
κ
to
M
k
,
via
the
Kummer
structure
of
(X
ell
M
k
),
the
two
one-parameter
subgroups
“S
1
×
{1},
{1}
×
R
>0
⊆
C
×
”,
we
obtain,
in
a
natural
way,
an
hol
.
object
of
C
TB
In
particular,
we
obtain
a
[1-]commutative
diagram
of
natural
functors
hol
C
TH
⏐
⏐
−→
hol
C
TB
hol
−→
C
TB
hol
C
TH
⏐
⏐
−→
EA
⏐
⏐
−→
TM
—
in
which
the
vertical
arrows
are
“mono-analyticization
functors”
[cf.
the
mono-
analyticization
functors
of
(ii);
the
construction
implicit
in
(ii),
(c)];
the
arrows
hol
hol
→
EA,
C
TB
→
TM
are
the
natural
projection
functors.
Let
us
write
C
TH
N
w
def
hol
=
Orb(C
TB
)
×
Orb(TM
),w
Th
[Z]
def
hol
)
×
Orb(TM
),w
Th
[Z]
N
w
=
Orb(C
TB
—
where
the
“,
w”
in
the
fibered
product
is
to
be
understood
as
referring
to
the
natural
functor
Th
[Z]
→
Orb(TM
)
given
by
the
assignment
“W
→
G
w
”
[cf.
(ii)].
Thus,
we
have
natural
functors
N
w
→
N
w
→
Th
[Z].
TOPICS
IN
ABSOLUTE
ANABELIAN
GEOMETRY
III
137
Remark
5.6.1.
(i)
Observe
that
a
monoid
M
may
be
thought
of
as
a
[1-]category
C
M
consisting
of
a
single
object
whose
monoid
of
endomorphisms
is
given
by
M
.
In
a
similar
vein,
a
ring
R,
whose
underlying
additive
group
we
denote
by
R
,
may
be
thought
of
as
a
2-category
consisting
of
the
single
1-category
C
R
,
together
with
the
functors
C
R
→
C
R
arising
from
left
multiplication
by
elements
of
R.
(ii)
The
constructions
of
(i)
suggest
that
whereas
a
monoid
may
be
thought
of
as
a
mathematical
object
with
“one
combinatorial
dimension”,
a
ring
should
be
thought
of
as
a
mathematical
object
with
“two
combinatorial
dimensions”.
More-
over,
in
the
case
of
an
MLF
k,
these
two
combinatorial
dimensions
may
be
thought
of
as
corresponding
to
the
two
cohomological
dimensions
of
the
absolute
Galois
group
of
k,
while
in
the
case
of
a
CAF
k,
these
two
combinatorial
dimensions
may
be
thought
of
as
corresponding
to
the
two
real
or
topological
dimensions
of
k.
Thus,
from
this
point
of
view,
it
is
natural
to
think
of
ring
structures
as
corresponding
to
holomorphic
structures
—
i.e.,
both
ring
and
holomorphic
structures
are
based
on
a
certain
complicated
“intertwining
of
the
underlying
two
combinatorial
dimensions”.
So
far,
in
the
theory
of
§1,
§2,
§3,
and
§4
of
the
present
paper,
the
emphasis
has
been
on
“holomorphic
structures”,
i.e.,
of
restricting
ourselves
to
sit-
uations
in
which
this
“complicated
intertwining”
is
rigid.
By
contrast,
the
various
ideas
introduced
in
Definition
5.6
relate
to
the
issue
of
disabling
this
rigidity
—
i.e.,
of
“passing
from
one
holomorphic
to
two
underlying
combinatorial/topological
dimensions”
—
an
operation
which,
as
was
discussed
in
Remarks
1.9.4,
2.7.3,
has
the
effect
of
leaving
only
one
of
the
two
combinatorial
dimensions
rigid.
Put
an-
other
way,
this
corresponds
to
the
operation
of
“passing
from
rings
to
monoids”;
this
is
the
principal
motivation
for
the
term
“mono-analyticization”.
The
following
result
is
elementary
and
well-known.
Proposition
5.7.
(Local
Volumes)
Let
k
be
either
a
mixed-characteristic
nonarchimedean
local
field
or
a
complex
archimedean
field.
(i)
Suppose
that
k
is
nonarchimedean
[cf.
Definition
3.1,
(i)].
Write
m
k
⊆
O
k
for
the
maximal
ideal
of
O
k
and
M(k)
for
the
set
of
compact
open
subsets
of
k.
Then:
(a)
There
exists
a
unique
map
μ
k
:
M(k)
→
R
>0
that
satisfies
the
following
properties:
(1)
additivity,
i.e.,
μ
k
(A
B)
=
μ
k
(A)
+
μ
k
(B),
for
A,
B
∈
M(k)
such
that
A
B
=
∅;
(2)
-translation
invariance,
i.e.,
μ
k
(A
+
x)
=
μ
k
(A),
for
A
∈
M(k),
x
∈
k;
(3)
normal-
ization,
i.e.,
μ
k
(O
k
)
=
1.
We
shall
refer
to
μ
k
(−)
as
the
volume
on
k.
def
Also,
we
shall
write
μ
log
k
(−)
=
log(μ
k
(−))
[where
log
denotes
the
natural
logarithm
R
>0
→
R]
and
refer
to
μ
log
k
(−)
as
the
log-volume
on
k.
If
the
residue
field
of
k
is
of
cardinality
p
f
,
where
p
is
a
prime
number
and
f
n
a
positive
integer,
then,
for
n
∈
Z,
μ
log
k
(m
k
)
=
−f
·
n
·
log(p).
138
SHINICHI
MOCHIZUKI
def
def
(b)
Let
x
∈
k
×
;
set
μ̇
k
(x)
=
μ
k
(x
·
O
k
),
μ̇
log
k
(x)
=
log(
μ̇
k
(x)).
Then
for
log
log
×
A
∈
M(k),
we
have
μ
k
(x·A)
=
μ
k
(A)+
μ̇
log
k
(x);
in
particular,
if
x
∈
O
k
,
log
then
μ
log
k
(x
·
A)
=
μ
k
(A).
(c)
Write
log
k
:
O
k
×
→
k
for
the
[p-adic]
logarithm
on
k.
Let
A
⊆
O
k
×
be
∼
an
open
subset
such
that
log
k
induces
a
bijection
A
→
log
k
(A).
Then
log
μ
log
k
(A)
=
μ
k
(log
k
(A)).
(ii)
Suppose
that
k
is
archimedean
[cf.
Definition
4.1,
(i)];
thus,
we
have
a
natural
decomposition
k
×
∼
=
O
k
×
×
R
>0
,
where
O
k
×
∼
=
S
1
,
and
we
note
that
the
projection
k
×
→
R
>0
extends
to
a
continuous
map
pr
R
:
k
→
R.
Write
M(k)
(respectively,
M̆(k))
for
the
set
of
nonempty
compact
subsets
A
⊆
k
(respectively,
A
⊆
k
×
)
such
that
A
projects
to
a
[compact]
subset
of
R
(respectively,
O
k
×
)
which
is
the
closure
of
its
interior
in
R
(respectively,
O
k
×
).
Then:
(a)
The
standard
R-valued
absolute
value
on
k
determines
a
Riemannian
metric
[as
well
as
a
Kähler
metric]
on
k
that
restricts
to
Riemannian
met-
∼
∼
rics
on
O
k
×
→
O
k
×
×
{1}
→
k
×
and
R
>0
→
{1}
×
R
>0
→
k
×
.
Integrating
these
metrics
over
the
projection
of
A
∈
M(k)
(respectively,
A
∈
M̆(k))
to
R
(respectively,
O
k
×
)
[i.e.,
“computing
the
length
of
A
relative
to
these
metrics”]
yields
a
map
μ
k
:
M(k)
→
R
>0
(respectively,
μ̆
k
:
M̆(k)
→
R
>0
)
that
satisfies
the
following
properties:
(1)
additivity,
i.e.,
μ
k
(A
B)
=
μ
k
(A)
+
μ
k
(B)
(respectively,
μ̆
k
(A
B)
=
μ̆
k
(A)
+
μ̆
k
(B)),
for
A,
B
∈
M(k)
(respectively,
A,
B
∈
M̆(k))
whose
projections
to
R
(respectively,
O
k
×
)
are
disjoint;
(2)
normalization,
i.e.,
μ
k
(O
k
)
=
1
(respectively,
μ̆
k
(O
k
×
)
=
2π).
We
shall
refer
to
μ
k
(−)
(respectively,
μ̆
k
(−))
as
the
ra-
dial
volume
(respectively,
angular
volume)
on
k.
Also,
we
shall
write
def
def
log
μ
log
k
(−)
=
log(μ
k
(−))
(respectively,
μ̆
k
(−)
=
log(μ̆
k
(−)))
and
refer
to
log
μ
log
k
(−)
(respectively,
μ̆
k
(−))
as
the
radial
log-volume
(respectively,
angular
log-volume)
on
k.
def
def
(b)
Let
x
∈
k
×
;
set
μ̇
k
(x)
=
μ
k
(x
·
O
k
),
μ̇
log
k
(x)
=
log(
μ̇
k
(x)).
Then
for
log
log
A
∈
M(k)
(respectively,
A
∈
M̆(k)),
we
have
μ
log
k
(x·A)
=
μ
k
(A)+
μ̇
k
(x)
log
log
×
(respectively,
μ̆
log
k
(x
·
A)
=
μ̆
k
(A));
in
particular,
if
x
∈
O
k
,
then
μ
k
(x
·
A)
=
μ
log
k
(A).
(c)
Write
exp
k
:
k
→
k
×
for
the
exponential
map
on
k.
Let
A
∈
M(k)
be
such
that
exp
k
(A)
⊆
O
k
×
,
and,
moreover,
the
maps
pr
R
and
exp
k
induce
∼
∼
log
bijections
A
→
pr
R
(A),
A
→
exp
k
(A).
Then
μ
log
k
(A)
=
μ̆
k
(exp
k
(A)).
TOPICS
IN
ABSOLUTE
ANABELIAN
GEOMETRY
III
139
Proof.
First,
we
consider
assertion
(i).
Part
(a)
follows
immediately
from
well-
known
properties
of
the
Haar
measure
on
the
locally
compact
[additive]
group
k.
Part
(b)
follows
immediately
from
the
uniqueness
portion
of
part
(a).
To
verify
part
(c)
for
arbitrary
A,
it
suffices
[by
the
additivity
property
of
μ
k
(−)]
to
verify
part
(c)
for
A
of
the
form
x
+
m
nk
for
n
a
sufficiently
large
positive
integer,
x
∈
O
k
×
.
∼
But
then
log
k
determines
a
bijection
x
+
m
nk
→
log
k
(x)
+
m
nk
,
so
the
equality
log
log
μ
log
k
(A)
=
μ
k
(log
k
(A))
follows
from
the
-translation
invariance
of
μ
k
(−).
This
completes
the
proof
of
assertion
(i).
Assertion
(ii)
follows
immediately
from
well-
known
properties
of
the
geometry
of
the
complex
plane.
Remark
5.7.1.
The
“log-compatibility”
[i.e.,
part
(c)]
of
Proposition
5.7,
(i),
(ii),
may
be
regarded
as
a
sort
of
“integrated
version”
of
the
fact
that
the
derivative
of
the
formal
power
series
log(1
+
X)
=
X
+
.
.
.
at
X
=
0
is
equal
to
1.
Moreover,
the
opposite
directions
of
the
“arrows
involved”
[i.e.,
logarithm
versus
exponential]
in
the
nonarchimedean
and
archimedean
cases
is
reminiscent
of
the
discussion
of
Remark
4.5.2.
Proposition
5.8.
(Mono-analytic
Reconstruction
of
Log-shells)
(i)
Let
G
k
be
the
absolute
Galois
group
of
an
MLF
k.
Then
there
exists
a
functorial
[i.e.,
relative
to
TG
]
“group-theoretic”
algorithm
for
constructing
×
ab
the
images
of
the
embeddings
O
k
→
G
ab
k
,
k
→
G
k
of
local
class
field
theory
[cf.
[Mzk9],
Proposition
1.2.1,
(iii),
(iv)].
Here,
the
asserted
“functoriality”
is
con-
travariant
and
induced
by
the
Verlagerung,
or
transfer,
map
on
abelianizations.
In
particular,
we
obtain
a
functorial
“group-theoretic”
algorithm
for
reconstructing
the
residue
characteristic
p
[cf.
[Mzk9],
Proposition
1.2.1,
(i)],
the
invariant
p
∗
[i.e.,
p
if
p
is
odd;
p
2
if
p
is
even
—
cf.
Definition
5.4,
(iii)],
the
cardinality
p
f
of
the
residue
field
of
k
[i.e.,
by
adding
1
to
the
cardinality
of
the
prime-to-p
torsion
of
k
×
],
the
absolute
degree
[k
:
Q
p
]
[i.e.,
as
the
dimension
of
O
k
×
⊗
Q
p
over
Q
p
],
the
absolute
ramification
index
e
=
[k
:
Q
p
]/f
,
and
the
order
p
m
of
the
subgroup
of
p-th
power
roots
of
unity
of
k
×
.
(ii)
The
algorithms
of
(i)
yield
a
functorial
[i.e.,
relative
to
TG
]
“group-
theoretic”
algorithm
“Ob(TG
)
G
→
Γ
×
non
(G)”
for
constructing
from
G
the
MLF
Γ
×
[cf.
§0]
non
-diagram
in
C
TS
×
O
×
(G)
k
⏐
⏐
→
k
(G)
⏐
⏐
k
∼
(G)
→
(k
)
pf
(G)
×
determined
by
the
diagram
of
Definition
5.4,
(iii),
hence
also
the
log-shell
I(G)
⊆
k
∼
(G)
of
Γ
×
non
(G).
(iii)
The
algorithms
of
(i)
yield
a
functorial
[i.e.,
relative
to
TG
]
“group-
theoretic”
algorithm
“Ob(TG
)
G
→
R
non
(G)”
for
constructing
from
G
the
topological
group
[which
is
isomorphic
to
R]
×
R
non
(G)
=
(k
(G)/O
×
(G))
∧
def
k
140
SHINICHI
MOCHIZUKI
—
where
“∧”
stands
for
the
completion
with
respect
to
the
order
structure
deter-
mined
by
the
nonnegative
elements,
i.e.,
the
image
of
O
(G)/O
×
(G)
—
equipped
k
k
with
a
distinguished
element,
namely,
the
“Frobenius
element”
F(G)
∈
R
non
(G)
[cf.
[Mzk9],
Proposition
1.2.1,
(iv)],
which
we
think
of
as
corresponding
to
the
el-
ement
f
G
·
log(p
G
)
∈
R,
where
p
G
,
f
G
are
the
invariants
“p”,
“f
”
of
(i).
Finally,
these
algorithms
also
yield
a
functorial,
“group-theoretic”
algorithm
for
constructing
the
log-volume
map
μ
log
(G)
:
M(k
∼
(G)
G
)
→
R
non
(G)
—
where
“M(−)”
is
as
in
Proposition
5.7,
(i);
the
superscript
“G”
denotes
the
submodule
of
G-invariants;
if
we
write
m
G
,
e
G
,
p
∗
G
for
the
invariants
“m”,
“e”,
“p
∗
”
of
(i),
then
one
may
think
of
μ
log
(G)
as
being
normalized
via
the
formula
μ
log
(G)(I(G))
=
{−1
−
m
G
/f
G
+
e
G
·
log(p
∗
G
)/log(p
G
)}
·
F(G)
—
determined
by
composing
the
map
μ
log
of
Proposition
5.7,
(i),
(a),
with
the
k
∼
(G)
G
∼
isomorphism
R
→
R
non
(G)
given
by
f
G
·log(p
G
)
→
F(G).
That
is
to
say,
“μ
log
(G)”
and
“M(k
∼
(G)
G
)”
are
well-defined
despite
the
fact
one
does
not
have
an
algorithm
for
reconstructing
the
field
structure
on
k
∼
(G)
G
[i.e.,
unlike
the
situation
discussed
in
Proposition
5.7,
(i)].
−
→
(iv)
Let
G
=
(C,
C
)
∈
Ob(TM
);
write
C
∼
→
C
×
for
the
[pointed]
universal
covering
of
C
×
[cf.
the
definition
of
“k
∼
k
×
”
in
Definition
4.1,
(i)];
thus,
we
regard
C
∼
as
a
topological
group
[isomorphic
to
R].
Then
the
evident
isomorphism
def
def
Lie
±
(C
∼
)
∼
=
Lie
±
(C
×
)
allows
one
to
regard
k
∼
(G)
=
C
∼
×C
∼
,
k
×
(G)
=
C
×
×C
∼
as
objects
of
TB.
Write
Seg(G)
for
the
equivalence
classes
of
compact
line
segments
on
C
∼
[i.e.,
compact
subsets
which
are
either
equal
to
the
closure
of
a
connected
open
set
or
are
of
cardinality
one],
relative
to
the
equivalence
relation
determined
by
translation
on
C
∼
.
Then
forming
the
union
of
two
compact
line
segments
whose
intersection
is
of
cardinality
one
determines
a
monoid
structure
∼
on
Seg(G)
with
respect
to
which
Seg(G)
→
R
≥0
.
In
particular,
this
monoid
structure
determines
a
structure
of
topological
monoid
on
Seg(G).
(v)
The
constructions
of
(iv)
yield
a
functorial
[i.e.,
relative
to
TM
]
algo-
×
rithm
“Ob(TM
)
G
→
Γ
×
arc
(G)”
for
constructing
from
G
the
Γ
arc
-diagram
in
hol
[cf.
§0]
C
TB
k
∼
(G)
=
C
∼
×
C
∼
k
×
(G)
=
C
×
×
C
∼
determined
by
the
diagram
of
Definition
5.4,
(v),
hence
also
the
log-shell
I(G)
=
{(a
·
x,
b
·
x)
|
x
∈
I
C
∼
;
a,
b
∈
R;
a
2
+
b
2
=
1}
⊆
k
∼
(G)
def
—
where
we
write
I
C
∼
⊆
C
∼
for
the
unique
compact
line
segment
on
C
∼
that
is
invariant
with
respect
to
the
action
of
±1
and,
moreover,
maps
bijectively,
except
for
its
endpoints,
to
C
×
—
of
Γ
×
arc
(G).
TOPICS
IN
ABSOLUTE
ANABELIAN
GEOMETRY
III
141
(vi)
The
constructions
of
(iv)
yield
a
functorial
[i.e.,
relative
to
TM
]
al-
G
→
R
arc
(G)”
for
constructing
from
G
the
topological
gorithm
“Ob(TM
)
group
[which
is
isomorphic
to
R]
def
R
arc
(G)
=
Seg(G)
gp
equipped
with
a
distinguished
element,
namely,
the
“Frobenius
element”
F(G)
∈
Seg(G)
⊆
R
arc
(G)
determined
by
a
compact
line
segment
that
maps
bijectively,
except
for
its
endpoints,
to
C
×
;
we
shall
think
of
F(G)
as
corresponding
to
2π
∈
R.
Finally,
these
algorithms
also
yield
a
functorial,
algorithm
for
constructing
the
radial
and
angular
log-volume
maps
μ
log
(G)
:
M(k
∼
(G))
→
R
arc
(G);
μ̆
log
(G)
:
M̆(k
∼
(G))
→
R
arc
(G)
—
where
“M(−)”,
“
M̆(−)”
are
as
in
Proposition
5.7,
(ii);
if,
in
the
style
of
the
definition
of
I(G)
in
(v),
we
write
∂I
C
∼
for
the
boundary
[i.e.,
the
two
endpoints]
of
I
C
∼
and
O
k
×
∼
(G)
=
{(a
·
x,
b
·
x)
|
x
∈
∂I
C
∼
;
a,
b
∈
R;
a
2
+
b
2
=
π
−2
}
⊆
k
∼
(G)
def
∼
[so
one
has
a
natural
bijection
R
>0
×
O
k
×
∼
(G)
→
k
∼
(G)\{0}],
then
one
may
think
of
μ
log
(G),
μ̆
log
(G)
as
being
normalized
via
the
formulas
μ
log
(G)(I(G))
=
μ̆
log
(G)(O
k
×
∼
(G))
−
log(2)
·
F(G)/2π
=
log(π)
·
F(G)/2π
log
—
determined
by
composing
the
maps
μ
log
k
∼
(G)
,
μ̆
k
∼
(G)
of
Proposition
5.7,
(ii),
(a),
∼
with
the
isomorphism
R
→
R
arc
(G)
given
by
2π
→
F(G).
That
is
to
say,
“μ
log
(G)”,
“μ̆
log
(G)”,
“M(k
∼
(G))”,
and
“
M̆(k
∼
(G))”
are
well-defined
despite
the
fact
one
does
not
have
an
algorithm
for
reconstructing
the
field
structure
on
k
∼
(G)
[i.e.,
unlike
the
situation
discussed
in
Proposition
5.7,
(ii)].
(vii)
Let
Z
be
an
elliptically
admissible
hyperbolic
orbicurve
over
an
alge-
braic
closure
of
Q;
V
∈
Ob(Th
[Z])
[cf.
the
notation
of
Definition
5.1,
(iv);
Definition
5.4,
(i)];
W
∈
Ob(Th
[Z])
the
mono-analyticization
of
V
[cf.
the
notation
of
Definition
5.6,
(ii)];
w
∈
W
non
(respectively,
w
∈
W
arc
).
Write
An
[N
w
]
for
the
category
whose
objects
consist
of
an
object
of
Th
[Z],
together
with
the
MLF
×
hol
×
object
of
Orb(C
TS
[
Γ
non
])
(respectively,
Orb(C
TB
[
Γ
arc
]))
given
by
applying
the
×
algorithm
“G
→
Γ
non
(G)”
of
(ii)
(respectively,
“G
→
Γ
×
arc
(G)”
of
(v))
to
the
object
of
Orb(TG
)
(respectively,
Orb(TM
))
obtained
by
projecting
[at
w
—
cf.
Defini-
tion
5.6,
(ii)]
the
given
object
of
Th
[Z],
and
whose
morphisms
are
the
morphisms
induced
by
Th
[Z].
Thus
we
obtain
a
natural
equivalence
of
categories
∼
Th
[Z]
→
An
[N
w
]
together
with
a
“forgetful
functor”
An
ψ
w,ν
:
An
[N
w
]
→
N
w
142
SHINICHI
MOCHIZUKI
def
×
[cf.
Definition
5.6,
(iii),
(iv)]
for
each
vertex
ν
of
Γ
×
w
=
Γ
non
(respectively,
def
×
Γ
×
w
=
Γ
arc
),
and
a
natural
transformation
An
ι
w,
An
An
:
ψ
w,ν
→
ψ
w,ν
1
2
for
each
edge
of
Γ
×
w
running
from
a
vertex
ν
1
to
a
vertex
ν
2
.
Finally,
we
shall
omit
the
symbol
“”
from
the
above
notation
to
denote
the
result
of
composing
the
functors
and
natural
transformations
discussed
above
with
the
natural
functor
N
w
→
N
w
;
also,
we
shall
replace
the
symbol
“An
”
by
the
symbol
“”
in
the
superscripts
of
the
above
notation
to
denote
the
result
of
restricting
the
functors
and
natural
transformations
discussed
above
to
Th
[Z].
Proof.
The
various
assertions
of
Proposition
5.8
are
immediate
from
the
definitions
and
the
references
quoted
in
these
definitions.
Remark
5.8.1.
(i)
One
way
to
summarize
the
archimedean
portion
of
Proposition
5.8
is
as
follows:
Suppose
that
one
starts
with
the
[Aut-]holomorphic
monoid
given
by
an
isomorph
of
O
C
[i.e.,
where
one
thinks
of
the
[Aut-]holomorphic
structure
on
O
C
as
consisting
of
a(n)
[Aut-]holomorphic
structure
on
(O
C
)
gp
=
C
×
]
arising
as
the
O
A
X
for
some
X
∈
Ob(EA).
The
operation
of
mono-analyticization
consists
of
“forgetting
the
rigidification
of
the
[Aut-]holomorphic
structure
furnished
by
X”
[cf.
Remark
2.7.3].
Thus,
applying
the
operation
of
mono-analyticization
to
an
isomorph
of
O
C
yields
the
object
of
TM
consisting
of
an
isomorph
of
the
topological
monoid
O
C
equipped
with
the
submonoid
corresponding
to
O
C
∩
R
>0
,
which
is
non-rigid,
in
the
sense
that
it
is
subject
to
dilations
[cf.
Remark
2.7.3].
On
the
other
hand:
From
the
point
of
view
of
the
theory
of
log-shells,
one
wishes
to
per-
form
the
operation
of
mono-analyticization
—
i.e.,
of
“forgetting
the
[Aut-
]holomorphic
structure”
—
in
such
a
way
that
one
does
not
obliterate
the
metric
rigidity
[i.e.,
the
“applicability”
of
the
theory
of
Proposition
5.7]
of
the
log-shells
involved.
This
is
precisely
what
is
achieved
by
the
use
of
the
category
TB
—
cf.,
especially,
hol
hol
→
C
TB
in
Definition
5.6,
(iv);
the
the
construction
of
the
natural
functor
C
TH
constructions
of
Proposition
5.8,
(iv),
(v),
(vi).
That
is
to
say,
the
“metric
rigidity”
of
log-shells
is
preserved
even
after
mono-analyticization
by
thinking
of
the
“metric
rigidity”
of
the
original
[Aut-]holomorphic
O
C
as
being
constituted
by
“the
metric
rigidity
of
S
1
∼
=
O
C
×
,
together
with
the
rotation
automor-
phisms
of
Lie(C
×
)
of
order
4”
[cf.
Definition
5.6,
(iv)].
That
is
to
say,
this
approach
to
describing
“[Aut-]holomorphic
metric
rigidity”
has
the
advantange
of
being
“immune
to
mono-analyticization”
—
cf.
the
construction
of
k
∼
(G)
as
“C
∼
×
C
∼
”
in
Proposition
5.8,
(iv).
On
the
other
hand,
it
has
the
TOPICS
IN
ABSOLUTE
ANABELIAN
GEOMETRY
III
143
disadvantage
that
it
is
not
compatible
[as
one
might
expect
from
any
sort
of
mono-
analyticization
operation!]
with
preserving
the
complex
archimedean
field
structure
of
“k
∼
”.
That
is
to
say,
the
two
factors
of
C
∼
appearing
in
the
product
“C
∼
×
C
∼
”
—
which
should
correspond
to
the
imaginary
and
real
portions
of
such
a
complex
archimedean
field
structure
—
may
only
be
related
to
one
another
up
to
a
{±1}
indeterminacy,
an
indeterminacy
that
has
the
effect
of
obliterating
the
ring/field
structure
involved.
(ii)
It
is
interesting
to
note
that
the
discussion
of
the
archimedean
situation
of
(i)
is
strongly
reminiscent
of
the
nonarchimedean
portion
of
Proposition
5.8,
which
allows
one
to
construct
metrically
rigid
log-shells
which
are
immune
to
mono-analyticization,
but
only
at
the
expense
of
sacrificing
the
ring/field
structures
involved.
Definition
5.9.
(i)
By
pulling
back
the
various
functorial
algorithms
of
Proposition
5.8
defined
on
TG
,
TM
via
the
mono-analyticization
functors
TG
sB
→
TG
,
EA
→
TM
,
we
obtain
functorial
algorithms
defined
on
TG
sB
,
EA.
In
particular,
if,
in
the
notation
of
Definition
5.1,
(iii)
(respectively,
Definition
5.1,
(iv);
Definition
5.6,
(ii)),
V
(respectively,
V
;
W
)
is
a
global
(respectively,
panalocal;
mono-analytic)
Galois-
def
theater,
then
for
each
v
∈
V
=
V
/Aut(Π)
(respectively,
v
∈
V
;
v
∈
W
),
we
obtain
—
i.e.,
by
applying
the
functorial
algorithms
“R
non
(−)”,
“R
arc
(−)”
of
Proposition
5.8,
(iii),
(vi)
—
[orbi-]topological
groups
[isomorphic
to
R]
R
v
equipped
with
distinguished
Frobenius
elements
F
v
∈
R
v
.
Moreover,
if
we
write
V
def
for
the
unique
global
element
of
V
=
V
/Aut(Π)
(respectively,
V
;
W
),
then
we
obtain
a(n)
[orbi-]topological
group
[isomorphic
to
R]
R
V
⊆
R
v
v
—
where
the
product
ranges
over
v
∈
V
(respectively,
v
∈
V
;
v
∈
W
)
—
obtained
as
the
“graph”
of
the
correspondences
between
the
R
v
’s
that
relate
the
F
v
/(f
v
·
log(p
v
))
[where
“f
v
”,
“p
v
”
are
the
invariants
“f
G
”,
“p
G
”
of
Proposition
5.8,
(iii)]
for
nonarchimedean
v
to
the
F
v
/2π
for
archimedean
v.
Thus,
R
V
is
equipped
with
a
distinguished
element
F
V
∈
R
V
[which
we
think
of
as
corresponding
to
1
∈
R],
∼
and
we
have
natural
isomorphisms
of
[orbi-]topological
groups
R
V
→
R
v
that
map
F
V
→
F
v
/(f
v
·
log(p
v
))
for
nonarchimedean
v
and
F
V
→
F
v
/2π
for
archimedean
v
[where
we
note
that
division
of
elements
of
the
abstract
topological
group
R
v
by
a
positive
real
number
is
well-defined].
(ii)
In
the
notation
of
Definition
5.1,
(v)
(respectively,
Definition
5.1,
(vi)),
let
M
(respectively,
M
)
be
a
global
(respectively,
panalocal)
T-pair,
for
T
∈
def
{TF,
TM}.
In
the
non-resp’d
case,
write
V
=
V
/Aut(Π).
Then
the
various
log-volumes
defined
in
Proposition
5.7,
(i),
(ii),
determine
maps
Π
v
{μ
log
v
:
M(M
v
)
→
R
v
}
v∈V
;
Π
v
{μ̆
log
v
:
M̆(M
v
)
→
R
v
}
v∈V
arc
144
SHINICHI
MOCHIZUKI
—
where
we
write
M(M
v
Π
v
),
[when
v
∈
V
arc
]
M̆(M
v
Π
v
=
M
v
)
for
the
set
of
sub-
sets
determined
[via
the
reference
isomorphisms
“ψ
v
”
of
Definition
5.1,
(v);
the
“forgetful
functors”
of
Corollary
5.2,
(iv),
(vii)]
by
intersecting
with
M
T
(Π,
v)
Π
v
⊆
k
NF
(Π,
v)
Π
v
the
corresponding
collection
of
subsets
of
M(k
NF
(Π,
v)
Π
v
),
[when
v
∈
×
V
arc
]
M̆(k
NF
(Π,
v)
Π
v
).
(iii)
In
the
non-resp’d
[i.e.,
global]
case
of
(ii),
suppose
further
that
T
=
TF.
Then
for
any
-line
bundle
L
on
M
,
one
verifies
immediately
that
there
exist
morphisms
of
-line
bundles
on
M
ζ
:
L
1
→
L
;
ζ
0
:
L
1
→
L
0
such
that
L
0
is
isomorphic
to
the
trivial
-line
bundle.
Thus,
for
each
v
∈
∼
V
,
we
obtain
isomorphisms
of
M
v
Π
v
-vector
spaces
ζ[v]
:
L
1
[v]
→
L
[v],
ζ
0
[v]
:
∼
L
1
[v]
→
L
0
[v].
Moreover,
by
applying
these
isomorphisms,
we
obtain
subsets
S
v
⊆
L
0
[v]
for
each
v
∈
V
as
follows:
If
v
∈
V
non
,
then
we
take
S
v
to
be
the
subset
determined
by
the
closure
of
the
image
[via
the
various
ρ
v
,
for
v
∈
V
lying
over
v]
of
L
[
].
If
v
∈
V
arc
,
then
we
take
S
v
to
be
the
subset
determined
by
the
set
of
elements
of
L
[v]
for
which
|
−
|
L
[v]
≤
1.
Now
set
def
mod
mod
μ
log
2μ
log
+
μ
log
∈
R
V
(L
)
=
v
(S
v
)
/d
v
v
(S
v
)
/d
v
v∈V
arc
v∈V
non
—
where
d
mod
is
as
in
Definition
5.1,
(ii),
for
v
∈
V
∼
=
V(F
mod
);
the
superscript
v
∼
“
”
denotes
the
result
of
applying
the
natural
isomorphisms
R
V
→
R
v
of
(i);
we
note
that
the
sum
is
finite,
since
μ
log
v
(S
v
)
=
0
for
all
but
finitely
many
v
∈
V
.
As
is
well-known
[or
easily
verified!]
from
elementary
number
theory
—
i.e.,
the
so-
called
“product
formula”!
—
it
follows
immediately
that
[as
the
notation
suggests]
and,
in
particular,
is
“μ
log
(L
)”
depends
only
on
the
isomorphism
class
of
L
independent
of
the
choice
of
ζ,
ζ
0
.
Finally,
by
applying
the
equivalences
of
categories
of
Definition
5.3,
(ii),
(iii),
it
follows
immediately
that
we
may
extend
the
R
V
-
valued
function
[on
isomorphism
classes
of
-line
bundles
on
M
]
μ
log
(−)
to
a
function
that
is
also
defined
on
isomorphism
classes
of
-line
bundles
on
M
,
for
arbitrary
T
∈
{TF,
TM,
TLG}.
Remark
5.9.1.
Just
as
in
Remark
5.3.1,
one
may
define
—
in
the
style
of
Corollary
5.2
—
a
category
An
[Th
•
T
,
μ],
where
•
∈
{
,
},
whose
objects
are
data
of
the
form
•
M
•μ
T
(Π)
=
(M
T
(Π),
def
log
log
{(R
v
,
μ
log
v
(Π
v
)(−))}
v∈V
non
,
{(R
v
,
μ
v
(X
v
)(−),
μ̆
v
(X
v
)(−))}
v∈V
arc
)
—
where
the
“(Π
v
)’s”,
“(X
v
)’s”
preceding
the
“(−)’s”
are
to
be
understood
as
denoting
the
log-volumes
associated,
as
in
Definition
5.9,
(ii),
to
the
various
con-
stituent
data
of
M
•
T
(Π)
—
for
Π
∈
Ob(EA
),
and
whose
morphisms
are
the
mor-
phisms
induced
by
morphisms
of
EA
.
In
a
similar
vein,
by
combining
the
data
that
constitutes
an
object
of
An
[Th
T
]
with
the
data
log
log
{(R
v
,
μ
log
v
(Π
v
)(−))}
v∈V
,
{(R
v
,
μ
v
(X
v
)(−),
μ̆
v
(X
v
)(−))}
v∈V
arc
TOPICS
IN
ABSOLUTE
ANABELIAN
GEOMETRY
III
145
—
where
the
“(Π
v
)’s”,
“(X
v
)’s”
preceding
the
“(−)’s”
are
to
be
understood
as
denoting
the
log-volumes
associated,
as
in
Definition
5.9,
(ii),
to
the
various
con-
stituent
data
of
the
original
object
of
An
[Th
T
]
—
and
considering
the
morphisms
induced
by
morphisms
of
Th
,
we
obtain
a
category
An
[Th
T
,
μ].
Finally,
by
com-
bining
the
constructions
of
Definitions
5.3,
5.9;
Remark
5.3.1,
we
obtain
a
category
An
[Th
T
,
||,
μ]
whose
objects
are
data
of
the
form
|
|μ
M
T
|
|
def
(Π)
=
(M
T
(Π),
R
V
,
log
log
log
{μ
log
v
(Π)(−)
}
v∈V
non
,
{μ
v
(Π)(−)
,
μ̆
v
(Π)(−)
}
v∈V
arc
,
μ
(Π)(−))
—
where
the
“(Π)’s”
preceding
the
“(−)’s”
are
to
be
understood
as
denoting
the
log-volumes
associated,
as
in
Definition
5.9,
(ii),
(iii),
to
the
various
constituent
|
|
data
of
M
T
(Π)
—
for
Π
∈
Ob(EA
),
and
whose
morphisms
are
the
morphisms
induced
by
morphisms
of
EA
.
Then,
just
as
in
Corollary
5.2,
Remark
5.3.1,
one
obtains
sequences
of
natural
functors
EA
→
An
[Th
•
T
,
μ]
→
An
[Th
•
T
]
→
Th
•
T
→
Th
•
Th
→
An
[Th
T
,
μ]
→
An
[Th
T
]
→
Th
T
→
Th
EA
→
An
[Th
T
,
||,
μ]
→
An
[Th
T
,
||]
→
Th
T
→
EA
—
where
the
first
arrows
are
the
functors
arising
from
the
definitions
of
the
cat-
egories
“An
[−,
μ]”,
“An
[−,
μ]”;
with
the
exception
of
the
second
to
last
arrow
of
the
first
line
of
the
above
display
in
the
case
where
•
=
,
every
arrow
of
the
above
display
is
an
equivalences
of
categories
[cf.
Corollary
5.2,
(i),
(iv),
(v),
(vii);
Remark
5.3.1].
Remark
5.9.2.
The
significance
of
measuring
[log-]volumes
in
units
that
belong
to
the
copies
of
R
determined
by
“R
non
(−)”,
“R
arc
(−)”
lies
in
the
fact
that
such
measurements
may
compared
on
both
sides
of
the
“log-wall”,
as
well
as
in
a
fashion
compatible
with
the
operation
of
mono-analyticization
[cf.
the
discussion
of
Remark
3.7.7;
Corollary
5.10,
(ii),
(iv),
below].
We
are
now
ready
to
state
the
main
result
of
the
present
§5
[and,
indeed,
of
the
present
paper!].
Corollary
5.10.
(Fundamental
Properties
of
Log-shells)
In
the
notation
of
Corollary
5.5;
Proposition
5.8,
(vii),
write
E
def
=
Th
[Z];
An
[N
def
]
=
An
[N
v
]
v∈V(F
mod
)
—
where
the
product
is
a
fibered
product
of
categories
over
E
=
Th
[Z].
Consider
146
SHINICHI
MOCHIZUKI
the
diagram
of
categories
D
N
v
⏐
⏐
N
v
⏐
⏐
...
...
N
v
⏐
⏐
...
N
v
N
v
...
...
N
v
⏐
⏐
...
...
...
E
⏐
⏐
An
[N
⏐
⏐
]
E
—
where
we
regard
the
rows
of
D
as
being
indexed
by
the
integers
3,
4,
5,
6,
7
[relative
to
which
we
shall
use
the
notation
“D
≤n
”
—
cf.
Corollary
5.5];
the
ar-
rows
of
D
≤5
are
those
discussed
in
Definition
5.6,
(iii),
(iv);
the
arrows
of
the
rows
numbered
5,
6,
7
of
D
are
the
arrows
deterimined
by
the
equivalence
of
cat-
egories
of
Proposition
5.8,
(vii).
Note,
moreover,
that
we
have
a
natural
mono-
analyticization
morphism
[consisting
of
arrows
between
corresponding
vertices
belonging
to
rows
indexed
by
the
same
integer!]
of
diagrams
of
categories
•
→D
D
≥3
[cf.
Definition
5.4,
(iv),
(vi),
as
well
as
the
discussion,
involving
panalocalization
and
mono-analyticization
functors,
of
Corollary
5.5,
(vi);
Definition
5.6,
(ii),
(iii),
(iv)]
—
where
the
subscript
“≥
3”
refers
to
the
portion
involving
the
rows
numbered
3,
4,
5,
6,
7,
and
we
take
the
arrow
An
•
[X
]
→
An
[N
]
to
be
the
arrow
induced,
via
the
equivalence
of
categories
κ
An
•
of
Corollary
5.5
and
the
equivalence
of
categories
of
Proposition
5.8,
(vii),
by
the
mono-analyticization
functor
E
•
→
E
;
write
D
•
for
the
diagram
of
categories
obtained
by
gluing
D
•
,
D
via
this
mono-analyticization
morphism.
We
shall
refer
to
the
various
isomorphisms
between
composites
of
func-
•
→
D
tors
inherent
in
the
definition
of
the
mono-analyticization
morphism
D
≥3
[e.g.,
the
natural
isomorphisms
between
the
functors
associated
to
the
two
length
2
paths
N
v
→
N
v
→
N
v
,
N
v
→
N
v
→
N
v
,
where
v
∈
V(F
mod
),
in
the
third
and
fourth
rows
of
D
•
]
as
mono-analyticization
homotopies.
We
shall
refer
to
the
natural
transformation
“ι
v,
”
of
Corollary
5.5,
(iii),
as
a
shell-homotopy
[at
v]
if
is
a
shell-arrow
[cf.
Definition
5.4,
(iii),
(v)];
we
shall
refer
to
“ι
v,
”
as
a
log-
homotopy
[at
v]
if
the
initial
vertex
of
is
a
post-log
vertex.
If
v
∈
V(F
mod
)
non
(respectively,
v
∈
V(F
mod
)
arc
),
then
we
shall
refer
to
as
a
•-shell-container
struc-
ture
on
an
object
S
∈
Ob(N
v
)
the
datum
of
an
object
S
∈
Ob(X
)
together
with
an
∼
isomorphism
S
→
λ
v,ν
(S
),
where
ν
is
the
terminal
(respectively,
initial)
vertex
of
TOPICS
IN
ABSOLUTE
ANABELIAN
GEOMETRY
III
147
the
shell-arrow
of
Γ
×
v
;
an
object
of
N
v
equipped
with
a
•-shell-container
structure
will
be
referred
to
as
a
•-shell-container.
Note
that
the
shell-homotopies
deter-
mine
•-log-shells
“I”
[cf.
Definition
5.4,
(iii),
(v)]
inside
the
underlying
object
of
TS
(respectively,
TH)
determined
by
each
•-shell-container.
If
v
∈
V(F
mod
)
non
(respectively,
v
∈
V(F
mod
)
arc
),
and
S
is
an
object
of
N
v
or
N
v
,
then
we
shall
write
S
Gal
for
the
topological
submodule
of
Galois-invariants
of
(respectively,
the
topological
submodule
constituted
by)
the
underlying
object
of
TS
(respectively,
TH
or
TB)
determined
by
S.
(i)
(Finite
Log-volume)
Let
v
∈
V(F
mod
)
non
(respectively,
v
∈
V(F
mod
)
arc
).
For
each
•-shell-container
S
∈
Ob(N
v
),
S
Gal
is
equipped
with
a
well-defined
log-
volume
(respectively,
well-defined
radial
and
angular
log-volumes)
[cf.
Defini-
tion
5.9,
(ii)]
that
depend(s)
only
on
the
•-shell-container
structure
of
S.
Moreover,
the
•-log-shell
is
contained
in
S
Gal
and
[relative
to
these
log-volumes]
is
of
finite
log-volume
(respectively,
finite
radial
log-volume).
(ii)
(Log-Frobenius
Compatibility
of
Log-volumes)
For
v,
S
as
in
(i),
the
log-volume
(respectively,
radial
log-volume),
computed
“at
∈
L”,
is
com-
patible
[cf.
part
(c)
of
Proposition
5.7,
(i),
(ii)],
relative
to
the
relevant
log-
homotopy,
with
the
log-volume
(respectively,
angular
log-volume),
computed
“at
+
1
∈
L”.
(iii)
(Panalocalization)
The
log-volumes
of
(i),
as
well
as
the
construction
of
the
•-log-shells
from
the
various
shell-homotopies,
are
compatible
with
the
panalocalization
morphism
D
→
D
of
Corollary
5.5,
(vi).
(iv)
(Mono-analyticization)
If
v
∈
V(F
mod
)
non
(respectively,
v
∈
V(F
mod
)
arc
),
then
we
shall
refer
to
as
a
•-shell-container
structure
on
an
object
S
∈
Ob(N
v
)
the
datum
of
an
object
S
∈
Ob(X
)
together
with
an
isomorphism
from
S
to
the
image
in
N
v
of
λ
v,ν
(S
),
where
ν
is
the
terminal
(respectively,
initial)
vertex
of
equipped
with
a
•-shell-container
struc-
the
shell-arrow
of
Γ
×
v
;
an
object
of
N
v
ture
will
be
referred
to
as
a
•-shell-container.
Note
that
the
shell-homotopies
determine
•-log-shells
“I”
[cf.
Definition
5.4,
(iii),
(v)]
inside
each
•-shell-
container,
as
well
as
a
well-defined
log-volume
(respectively,
well-defined
radial
and
angular
log-volumes)
on
the
Gal-superscripted
module
associated
to
a
•-
shell-container
[cf.
(i)].
These
•-log-shells
and
log-volumes
depend
only
on
the
mono-analyticized
data
[i.e.,
roughly
speaking,
the
data
contained
in
D
],
in
the
following
sense
[cf.,
especially,
(d)]:
•
admits
a
natural
structure
(a)
(Mono-analytic
Cores)
For
n
=
5,
6,
7,
D
≤n
•
of
core
on
the
subdiagram
of
categories
of
D
determined
by
the
union
•
•
D
≤n
—
i.e.,
loosely
speaking,
E
,
An
[N
]
“form
cores”
of
the
D
≤n−1
•
functors
in
D
.
(b)
(Mono-analytic
Telecores)
As
v
ranges
over
the
elements
of
V(F
mod
)
and
ν
over
the
elements
of
Γ
×
v
,
the
restrictions
An
φ
v,ν
:
An
[N
]
→
N
v
An
to
An
[N
]
of
the
“forgetful
functors”
ψ
v,ν
of
Proposition
5.8,
•
•
D
≤6
,
whose
(vii),
give
rise
to
a
telecore
structure
T
An
on
D
≤5
148
SHINICHI
MOCHIZUKI
underlying
diagram
of
categories
we
denote
by
D
An
,
by
appending
to
•
An
telecore
edges
corresponding
to
the
arrows
φ
v,ν
from
the
core
D
≤6
An
[N
]
to
the
vertices
of
the
row
of
D
indexed
by
the
integer
3.
More-
over,
the
respective
family
of
homotopies
of
T
An
and
the
observables
S
log
,
S
log
of
Corollary
5.5,
(iii),
are
compatible.
(c)
(Mono-analytic
Contact
Structures)
For
v
∈
V(F
mod
),
ν
∈
Γ
×
v
,
there
is
a
natural
isomorphism
η
v,ν
from
the
composite
functor
deter-
1
[of
length
6]
mined
by
the
path
γ
v,ν
X
λ
v,ν
−→
N
v
−→
N
v
−→
E
•
⏐
⏐
E
−→
An
[N
]
An
φ
v,ν
−→
N
v
•
on
Γ
D
An
—
where
the
first
three
arrows
lie
in
D
≤5
,
the
fourth
arrow
•
arises
from
the
mono-analyticization
morphism
D
≥3
→
D
,
and
the
fifth
arrow
lies
in
D
—
to
the
composite
functor
determined
by
the
path
0
[of
length
2]
γ
v,ν
λ
v,ν
X
−→
N
v
−→
N
v
on
Γ
D
An
.
Moreover,
the
resulting
homotopies
η
v,ν
,
(η
v,ν
)
−1
,
together
with
the
mono-analyticization
homotopies
and
the
homotopies
on
”
[cf.
Proposition
5.8,
(vii)],
generate
a
D
An
arising
from
the
“ι
An
v,
contact
structure
H
An
on
T
An
that
is
compatible
with
the
telecore
and
contact
structures
T
An
•
,
H
An
•
of
Corollary
5.5,
(ii),
as
well
as
with
the
homotopies
of
the
observables
S
log
,
S
log
of
Corollary
5.5,
(iii),
that
log
arise
from
the
“ι
v,
”,
“ι
v,
”
indexed
by
∈
Γ
×
v
[not
Γ
v
!].
(d)
(Mono-analytic
Log-shells)
If
v
∈
V(F
mod
)
non
(respectively,
v
∈
V(F
mod
)
arc
),
then
we
shall
refer
to
as
a
-shell-container
structure
on
an
object
S
∈
Ob(N
v
)
the
datum
of
an
object
S
∈
Ob(An
[N
]),
∼
An
together
with
an
isomorphism
from
S
→
φ
v,ν
(S
),
where
ν
is
the
ter-
minal
(respectively,
initial)
vertex
of
the
shell-arrow
of
Γ
×
v
;
an
object
of
N
v
equipped
with
a
-shell-container
structure
will
be
referred
to
as
a
-shell-container.
Note
that
the
portion
of
the
data
that
constitutes
an
object
of
An
[N
v
]
determined
by
the
shell-arrow
gives
rise
to
a
-log-
shell
“I”
inside
each
-shell-container,
as
well
as
to
a
log-volume
on
the
Gal-superscripted
module
associated
to
a
-shell-container
[cf.
Proposition
5.8,
(ii),
(iii),
(v),
(vi)].
Finally,
every
•-shell-container
structure
on
an
object
of
N
v
determines,
by
applying
the
isomorphism
η
v,ν
of
(c),
a
corresponding
-shell-container
structure
on
the
object;
this
correspon-
dence
between
•-,
-shell-container
structures
is
compatible
with
the
•-,
-log-shells,
as
well
as
with
the
various
log-volumes,
determined,
respectively,
by
these
•-,
-shell-container
structures.
TOPICS
IN
ABSOLUTE
ANABELIAN
GEOMETRY
III
149
Proof.
The
various
assertions
of
Corollary
5.10
are
immediate
from
the
definitions,
together
with
the
references
quoted
in
the
statement
of
Corollary
5.10.
Here,
we
note
that
in
the
nonarchimedean
portion
of
part
(c)
of
assertion
(iv),
in
order
to
construct
the
isomorphisms
η
v,ν
,
it
is
necessary
to
relate
the
construction
of
the
base
field
as
a
subset
of
abelianizations
of
various
open
subgroups
of
the
Galois
group
[cf.
Proposition
5.8,
(i)]
to
the
Kummer-theoretic
construction
of
the
base
field
as
performed
in
Theorem
1.9,
(e).
This
may
be
achieved
by
applying
the
group-theoretic
construction
algorithms
of
Corollary
1.10
—
i.e.,
more
precisely,
by
combining
the
“fundamental
class”
natural
isomorphism
of
Corollary
1.10,
(a)
[cf.
also
the
first
isomorphism
of
the
display
of
Corollary
1.10,
(b)],
with
the
cyclotomic
natural
isomorphism
of
Corollary
1.10,
(c)
[cf.
also
Remark
1.10.3,
(ii)].
Put
another
way,
this
series
of
algorithms
may
be
summarized
as
a
“group-theoretic
algorithm
for
constructing
the
reciprocity
map
of
local
class
field
theory”.
Remark
5.10.1.
(i)
Note
that,
in
the
notation
of
Corollary
5.10,
(iv),
(c),
by
pre-composing
η
v,ν
with
the
telecore
arrow
φ
:
An
•
[X
]
→
X
of
Corollary
5.5,
(ii),
and
applying
the
coricity
of
Corollary
5.5,
(i),
together
with
an
appropriate
mono-analyticization
homotopy,
we
obtain
that
one
may
think
of
η
v,ν
as
yielding
a
homotopy
from
the
path
φ
An
v,ν
•
An
[X
]
−→
An
[N
]
−→
N
v
—
which
is
somewhat
simpler
[hence
perhaps
easier
to
grasp
intuitively]
than
the
domain
path
of
the
original
homotopy
η
v,ν
—
to
the
path
•
An
[X
]
φ
−→
X
λ
v,ν
−→
N
v
−→
N
v
0
[i.e.,
obtained
by
simply
pre-composing
γ
v,ν
with
φ
].
(ii)
Note
that
the
isomorphism
of
(i)
between
the
two
composites
of
functors
An
[X
]
→
N
v
depends
only
on
“Galois-theoretic/Aut-holomorphic
data”.
In
particular,
one
may
construct
—
in
the
style
of
Remarks
5.3.1,
5.9.1
—
a
category
“An
•
[X
,
η
]”
whose
objects
consist
of
the
data
of
objects
of
An
•
[X
],
together
with
the
algorithms
used
to
construct
the
various
homotopies
of
(i)
arising
from
η
v,ν
[i.e.,
associated
to
the
various
v
∈
V(F
mod
),
ν
∈
Γ
×
v
],
and
whose
morphisms
are
the
•
morphisms
induced
by
morphisms
of
An
[X
].
That
is
to
say,
objects
of
An
•
[X
,
η
]
consist
of
objects
of
An
•
[X
],
together
with
“group-theoretic
algorithms
encoding
the
reciprocity
law
of
local
class
field
theory
at
the
nonarchimedean
primes
and
the
archimedean
analogue
of
these
algorithms
at
the
archimedean
primes”.
Moreover,
the
“forgetful
functor”
∼
An
•
[X
,
η
]
→
An
•
[X
]
•
determines
a
natural
equivalence
of
categories.
Finally,
one
verifies
immediately
that
one
may
replace
“An
•
[X
]”
by
“An
•
[X
,
η
]”
in
Corollaries
5.5
and
5.10
with-
out
affecting
the
validity
of
their
content
—
e.g.,
without
affecting
the
coricity
of
Corollary
5.5,
(i).
We
leave
the
routine
details
to
the
interested
reader.
Remark
5.10.2.
The
significance
of
the
theory
of
log-shells
as
summarized
in
Corollary
5.10
—
and,
more
generally,
of
the
entire
theory
of
the
present
paper
—
may
be
understood
in
more
intuitive
terms
as
follows.
150
SHINICHI
MOCHIZUKI
(i)
One
important
aspect
of
the
classical
theory
of
line
bundles
on
a
proper
curve
[over
a
field]
is
that
although
such
line
bundles
exhibit
a
certain
rigidity
arising
from
the
properness
of
the
curve,
this
rigidity
is
obliterated
by
Zariski
localization
on
the
curve.
Put
another
way,
to
work
with
line
bundles
up
to
isomorphism
amounts
to
allowing
oneself
to
“multiply
the
line
bundle
by
a
rational
function”,
i.e.,
to
work
up
to
rational
equivalence.
Although
rational
equivalence
does
not
obliterate
the
global
degree
of
a
line
bundle
over
the
entire
proper
curve,
if
one
thinks
of
a
line
bundle
as
a
collection
of
integral
structures
at
the
various
primes
of
the
curve,
then
rational
equivalence
has
the
effect
of
“rearranging
these
integral
structures”
at
the
various
primes.
⎛
⎞
⎞
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
v
1
v
2
v
3
v
4
v
5
⎟
⎟
⎟
⎟
⎟
⎟
⎟
...
⎟
⎟
⎟
⎟
⎟
⎟
⎠
rational
equivalence
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
v
1
⎟
⎟
⎟
⎟
⎟
⎟
⎟
...
⎟
⎟
⎟
⎟
⎟
⎟
⎠
v
2
v
3
v
4
v
5
⎛
⎞
⎛
⎞
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
⎟
⎟
⎟
⎟
⎟
⎟
⎟
...
⎟
⎟
⎟
⎟
⎟
⎟
⎠
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
⎟
⎟
⎟
⎟
⎟
⎟
⎟
...
⎟
⎟
⎟
⎟
⎟
⎟
⎠
v
1
v
2
v
3
v
4
v
5
panalocalized
log-shells
v
1
v
2
v
3
v
4
v
5
If
one
restricts
oneself
to
working
globally
on
the
proper
curve,
then
such
“rear-
rangements”
are
coordinated
with
one
another
in
such
a
way
as
to
preserve,
for
instance,
the
global
degree;
on
the
other
hand,
if
one
further
imposes
the
condition
of
compatibility
with
Zariski
localization,
then
such
“coordination
of
integral
struc-
ture”
mechanisms
are
obliterated.
By
contrast,
the
“MF
∇
-objects”
of
[Falt]
satisfy
a
certain
“extraordinary
rigidity”
with
respect
to
Zariski
localization
that
reflects
the
fact
that
they
form
a
category
that
is
equivalent
to
a
certain
category
of
Galois
representations.
From
the
point
of
view
of
thinking
of
line
bundles
as
collections
of
integral
structures
at
the
various
primes,
the
rigidity
of
MF
∇
-objects
may
be
thought
of
as
a
sort
of
“freezing
of
the
integral
structures”
at
the
various
primes
in
a
fashion
that
is
immune
to
the
gluing
indeterminacies
that
occur
for
line
bundles
upon
execution
of
Zariski
localization
operations.
Put
another
way,
this
rigidity
may
be
thought
of
as
a
sort
of
“immunity
to
social
isolation”
from
other
primes.
In
the
context
of
Corollary
5.10,
this
property
corresponds
to
the
panalocalizability
[i.e.,
Corollary
5.10,
(iii)]
of
[the
integral
structures
constituted
by]
log-shells.
TOPICS
IN
ABSOLUTE
ANABELIAN
GEOMETRY
III
151
(ii)
At
this
point,
it
is
useful
to
observe
that,
at
least
from
an
a
priori
point
of
view,
there
exist
other
ways
in
which
one
might
attempt
to
“freeze
the
local
integral
structures”.
For
instance,
instead
of
working
strictly
with
line
bundles,
one
could
consider
the
ring
structure
of
the
global
ring
of
integers
of
a
number
field
[which
gives
rise
to
the
trivial
line
bundle
—
cf.
Definition
5.3,
(ii)];
that
is
to
say,
by
con-
sidering
ring
structures,
one
obtains
a
“rigid
integral
structure”
that
is
compatible
with
Zariski
localization
—
i.e.,
by
considering
the
ring
structure
“O”
of
the
local
rings
of
integers
[cf.
Remark
5.4.3].
Indeed,
log-shells
may
be
thought
of
—
and,
moreover,
were
originally
intended
by
the
author
—
as
a
sort
of
approximation
of
these
local
integral
structures
“O”
[cf.
Remark
5.4.2].
On
the
other
hand,
this
sort
of
rigidification
of
local
integral
structures
that
makes
essential
use
of
the
ring
structure
is
no
longer
compatible
with
the
operation
of
mono-analyticization
[cf.
Remark
5.6.1],
i.e.,
of
forgetting
one
of
the
two
combinatorial
dimensions
“”,
“”
that
constitute
the
ring
structure.
Thus,
another
crucial
property
of
log-shells
is
their
compatibility
with
mono-analyticization,
as
documented
in
Corollary
5.10,
(iv)
[cf.
also
Remarks
5.8.1,
5.9.2;
Definition
5.9,
(iii)],
i.e.,
their
“immunity
to
social
isolation”
from
the
given
ring
structures.
From
the
point
of
view
of
the
theory
of
§1,
§2,
§3,
§4,
such
ring
structures
may
be
thought
of
as
“arithmetic
holomorphic
structures”
[i.e.,
outer
Galois
actions
at
nonarchimedean
primes
and
Aut-holomorphic
structures
at
archimedean
primes]
—
cf.
Remark
5.6.1.
Thus,
if
one
thinks
of
the
result
of
forgetting
such
“arithmetic
holomorphic
structures”
as
being
like
a
sort
of
“arithmetic
real
analytic
core”
on
which
various
“arithmetic
holomorphic
structures”
may
be
imposed
—
i.e.,
a
sort
of
arithmetic
analogue
of
the
underlying
real
analytic
surface
of
a
Riemann
surface,
on
which
various
holo-
morphic
structures
may
be
imposed
[cf.
Remark
5.10.3
below]
—
then
the
theory
of
mono-analyticization
of
log-shells
guarantees
that
log-shells
remain
meaningful
even
as
one
travels
back
and
forth
between
various
“zones
of
arithmetic
holo-
morphy”
joined
—
in
a
fashion
reminiscent
of
spokes
emanating
from
a
core
—
by
a
single
“mono-analytic
core”.
...
arith.
hol.
str.
A
...
arith.
hol.
str.
B
mono-analytic
core
arith.
hol.
str.
C
...
arith.
hol.
str.
D
...
152
SHINICHI
MOCHIZUKI
(iii)
Another
approach
to
constructing
“mono-analytic
rigid
local
integral
struc-
tures”
is
to
work
with
the
local
monoids
“O
”
[i.e.,
as
opposed
to
“log(O
×
)”,
as
was
done
in
the
case
of
log-shells].
Here,
“O
”
may
be
thought
of
as
a
[possibly
twisted]
product
of
“O
×
”
with
some
“valuation
monoid”
that
consists
of
a
sub-
monoid
of
R
≥0
.
For
instance,
in
the
[complex]
archimedean
case,
O
C
∼
=
O
C
×
×
R
>0
.
On
the
other
hand
[cf.
Remark
5.6.1],
the
dimension
constituted
by
the
“valua-
tion
monoid”
R
>0
fails
to
retain
its
rigidity
when
subjected
to
the
operation
of
mono-analyticization.
The
resulting
“dilations”
of
R
>0
[i.e.,
by
raising
to
the
λ-
th
power,
for
λ
∈
R
>0
]
may
be
thought
of
as
being
like
Teichmüller
dilations
of
the
mono-analytic
core
discussed
in
(ii)
above
[cf.
also
the
discussion
of
Remark
5.10.3
below].
If,
moreover,
one
is
to
retain
a
coherent
theory
of
global
degrees
of
arithmetic
line
bundles
in
the
presence
of
such
“arithmetic
Teichmüller
dila-
tions”,
then
[in
order
to
preserve
the
“product
formula”
of
elementary
number
theory]
it
is
necessary
to
subject
the
valuation
monoids
at
nonarchimedean
primes
to
“arithmetic
Teichmüller
dilations”
which
are
“synchronized”
with
the
dilations
that
occur
at
the
archimedean
primes.
From
the
point
of
the
theory
of
Frobenioids
of
[Mzk16],
[Mzk17],
such
“arithmetic
Teichmüller
dilations”
at
nonarchimedean
primes
are
given
by
the
unit-linear
Frobenius
functor
studied
in
[Mzk16],
Proposi-
tion
2.5.
Thus,
in
summary:
In
order
to
guarantee
the
rigidity
of
the
local
integral
structures
under
consideration
when
subject
to
mono-analyticization,
one
must
abandon
the
“valuation
monoid”
portion
of
“O
”,
i.e.,
one
is
obliged
to
restrict
one’s
attention
to
the
“O
×
”
portion
of
“O
”.
On
the
other
hand,
within
each
zone
of
arithmetic
holomorphy
[cf.
(ii)],
one
wishes
to
consider
various
diverse
modifications
of
integral
structure
on
the
“rigid
standard
integral
structures”
that
one
constructs.
Since
this
is
not
possible
if
one
restricts
oneself
to
“O
×
”
regarded
multiplicatively,
one
is
thus
led
to
working
with
“log(O
×
)”
—
i.e.,
in
effect
with
the
log-shells
discussed
in
Corollary
5.10.
Thus,
within
each
zone
of
arithmetic
holomorphy,
one
wishes
to
convert
the
“”
operation
of
“O
×
”
into
a
“”
operation,
i.e.,
by
applying
the
logarithm.
On
the
other
hand,
when
one
leaves
that
zone
of
arithmetic
holomorphy,
one
wishes
to
return
again
to
work-
ing
with
“O
×
”
multiplicatively,
so
as
to
achieve
compatibility
with
the
operation
of
mono-analyticization.
Here,
we
note
that
-line
bundles
—
i.e.,
in
other
words,
line
bundles
regarded
from
an
idèlic
point
of
view
—
have
the
virtue
of
being
defined
using
only
the
multiplicative
structure
of
the
rings
involved
[cf.
the
theory
of
Frobe-
nioids
of
[Mzk16],
[Mzk17]],
hence
of
being
compatible
with
mono-analyticization.
[We
remark
here
that
the
detailed
specification
of
precisely
which
monoids
we
wish
to
use
when
we
apply
the
theory
of
Frobenioids
is
beyond
the
scope
of
the
present
paper.]
By
contrast,
although
-line
bundles
—
i.e.,
line
bundes
regarded
as
mod-
ules
of
a
certain
type
—
are
not
compatible
with
mono-analyticization,
they
have
the
virtue
of
allowing
us
to
relate,
within
each
zone
of
arithmetic
holomorphy,
the
additive
module
“log(O
×
)”
to
the
theory
of
-line
bundles
[which
is
compatible
with
mono-analyticization].
Thus,
in
summary:
This
state
of
affairs
obliges
one
to
work
in
a
“framework”
in
which
one
may
pass
freely,
within
each
zone
of
arithmetic
holomorphy,
back
and
forth
between
“”
and
“”
via
application
of
the
logarithm
at
the
various
nonarchimedean
and
archimedean
primes.
TOPICS
IN
ABSOLUTE
ANABELIAN
GEOMETRY
III
153
On
the
other
hand,
since
the
logarithm
is
not
a
ring
homomorphism,
it
is
not
at
all
clear,
a
priori,
how
to
establish
a
framework
in
which
one
may
apply
the
logarithm
at
will
[within
each
zone
of
arithmetic
holomorphy],
without
obliterating
the
foundations
[e.g.,
scheme-theoretic!]
underlying
the
mathematical
objects
that
one
works
with,
and,
moreover,
[a
related
issue
—
cf.
Remark
5.4.1]
without
obliterating
the
crucial
global
structure
of
the
number
fields
involved
[which
is
necessary
to
make
sense
of
global
arithmetic
line
bundles!].
A
solution
to
this
problem
of
finding
an
appropriate
“framework”
as
dis-
cussed
above
is
precisely
what
is
provided
by
“Galois
theory”
[cf.
also
the
“log-invariant
log-volumes”
of
Corollary
5.10,
(i),
(ii)]
—
which
is
both
global
and
“log-invariant”;
the
sufficiency
of
this
“framework”
[from
the
point
of
view
of
carrying
out
various
arithmetic
operations
involving
line
bundles,
as
discussed
above]
is
precisely
what
is
guaranteed
by
the
mono-
anabelian
theory
of
Corollaries
3.6,
4.5,
5.5.
log
log
core:
some
fixed
arithmetic
holomorphic
structure
log
!
log
...
At
a
more
philosophical
level,
the
“log-invariant
core”
furnished
by
“Galois
theory”
[cf.
the
remarks
concerning
telecores
following
Corollaries
3.6,
3.7]
and
supported,
in
content,
by
“mono-anabelian
geometry”
may
be
thought
of
as
a
“geometry
over
F
1
”
[i.e.,
over
the
fictitious
field
of
absolute
constants
in
Z]
with
respect
to
which
the
logarithm
is
“F
1
-linear”.
(iv)
Note
that
in
order
to
work
with
-line
bundles
[cf.
the
discussion
of
(iii);
Definition
5.3,
(ii)],
it
is
necessary
[unlike
the
case
with
-line
bundles]
to
work
with
all
the
primes
of
a
number
field.
Indeed,
to
work
with
“line
bundles”
in
a
fashion
that
allows
one
to
ignore
some
nonempty
set
of
primes
of
the
number
field
amounts
to
working
with
a
notion
of
rational
equivalence
that
involves
some
proper
subgroup
of
the
multiplicative
group
F
×
associated
to
the
number
field
F
.
On
the
other,
the
only
subgroups
of
F
×
that
[if
one
considers
the
union
of
F
×
with
{0}]
are
closed
under
addition
are
the
subgroups
of
F
×
that
arise
from
subfields
of
F
,
i.e.,
which
correspond,
in
effect,
to
-line
bundles
as
in
Definition
5.3,
(ii).
(v)
The
importance
of
the
process
of
mono-analyticization
in
the
discussion
of
(ii),
(iii)
is
reminiscent
of
the
discussion
in
[Mzk18],
Remark
1.10.4,
concerning
the
topic
of
“restricting
oneself
to
working
only
with
multiplicative
structures”
in
the
context
of
the
theory
of
the
étale
theta
function.
154
SHINICHI
MOCHIZUKI
(vi)
Finally,
we
recall
that
from
the
point
of
view
of
the
discussion
of
telecores
in
the
remarks
following
Corollaries
3.6,
3.7,
the
various
“forgetful
functors”
of
assertion
(ii)
of
Corollaries
3.6,
4.5,
5.5
may
be
thought
of
as
being
analogous
to
passing
to
the
“underlying
vector
bundle
plus
Hodge
filtration”
of
an
MF
∇
-object
[cf.
Remark
3.7.2].
From
this
point
of
view:
Log-shells
may
be
thought
of,
in
the
context
of
this
analogy
with
MF
∇
-
objects,
as
corresponding
to
the
section
of
a
[projective]
nilpotent
ad-
missible
indigenous
bundle
in
positive
characteristic
determined
by
the
p-curvature
[i.e.,
in
other
words,
the
Frobenius
conjugate
of
the
Hodge
filtration].
Remark
5.10.3.
From
the
point
of
the
view
of
the
analogy
of
the
theory
of
mono-
anabelian
log-compatibility
[cf.
§3,
§4]
with
the
theory
of
uniformizing
MF
∇
-objects
[cf.
Remark
3.7.2],
the
global/panalocal/mono-analytic
theory
of
log-shells
presented
in
the
present
§5
may
be
understood
as
follows.
(i)
The
mathematical
apparatus
on
a
number
field
arising
from
the
global/pana-
local
mono-anabelian
log-compatibility
of
Corollary
5.5
may
be
thought
of
as
being
analogous
to
the
[mod
p]
MF
∇
-object
constituted
by
a
nilpotent
indigenous
bundle
on
a
hyperbolic
curve
in
positive
characteristic
[cf.
the
theory
of
[Mzk1],
[Mzk4]].
Note
that
this
mathematical
apparatus
on
a
number
field
arises,
essentially,
from
the
outer
Galois
representation
determined
by
a
once-punctured
elliptic
curve
over
the
number
field.
That
is
to
say,
roughly
speaking,
we
have
correspondences
as
follows:
number
field
F
once-punctured
ell.
curve
X
over
F
←→
hyperbolic
curve
C
in
pos.
char.
←→
nilp.
indig.
bundle
P
over
C.
Here,
we
note
that
the
correspondence
between
number
fields
and
curves
over
finite
fields
is
quite
classical;
the
correspondence
between
families
of
elliptic
curves
and
indigenous
bundles
is
natural
in
the
sense
that
the
most
fundamental
example
of
an
indigenous
bundle
is
given
by
the
projectivization
of
the
first
de
Rham
cohomology
module
of
the
tautological
family
of
elliptic
curves
over
the
moduli
stack
of
elliptic
curves.
Note,
moreover,
that:
Just
as
in
the
case
of
indigenous
bundles,
the
fact
that
the
Kodaira-Spencer
morphism
is
an
isomorphism
may
be
interpreted
as
asserting
that
the
base
curve
“entrusts
its
local
moduli
to
the
indigenous
bundle”,
in
the
mono-anabelian
theory
of
the
present
paper,
the
various
localizations
of
a
number
field
“entrust
their
ring
structures
to
the
mono-anabelian
data
determined
by
the
once-punctured
elliptic
curve”
[cf.
Remarks
1.9.4,
2.7.3,
5.6.1;
Remark
5.10.2,
(iii)].
Relative
to
this
analogy,
we
observe
that
panalocalizability
corresponds
to
the
local
rigidity
of
MF
∇
-objects
[cf.
Remark
5.10.2,
(i)].
Moreover,
the
operation
of
mono-
analyticization
—
i.e.,
“forgetting
the
once-punctured
elliptic
curve”
—
corresponds
to
forgetting
the
indigenous
bundle,
hence
to
relinquishing
control
of
the
local
moduli
of
the
base
curve
C;
thus,
just
as
this
led
to
“Teichmüller
dilations”
in
the
discussion
of
Remark
5.10.2,
(ii),
(iii),
in
the
theory
of
indigenous
bundles,
forgetting
the
TOPICS
IN
ABSOLUTE
ANABELIAN
GEOMETRY
III
155
indigenous
bundle
means,
in
particular,
loss
of
control
of
the
deformation
moduli
of
the
base
curve
C.
Another
noteworthy
aspect
of
this
analogy
may
be
seen
in
the
fact
that:
Just
as
the
log-Frobenius
operation
only
exists
for
local
fields
[cf.
Re-
mark
5.4.1],
in
the
theory
of
indigenous
bundles,
Frobenius
liftings
only
exist
Zariski
locally
on
the
base
curve
C.
On
the
other
hand,
unlike
the
“linear
algebra-theoretic”
nature
of
the
theory
of
indigenous
bundles
[which
may
be
thought
of
as
sl
2
-bundles],
the
outer
Galois
representations
that
appear
in
the
theory
of
the
present
paper
are
fundamentally
“anabelian”
in
nature
—
i.e.,
their
“non-abelian
nature”
is
not
limited
to
a
rela-
tively
weak
“linear
algebra-theoretic”
departure
from
abelianity,
but
rather
on
a
par
with
that
of
[profinite]
free
groups.
In
particular,
unlike
the
linear
algebra-
theoretic
[i.e.,
“sl
2
-theoretic”]
nature
of
the
intertwining
of
the
two
dimensions
of
an
indigenous
vector
bundle,
the
two
combinatorial
dimensions
involved
[cf.
Re-
mark
5.6.1]
are
intertwined
in
an
essentially
anabelian
fashion
[i.e.,
constitute
a
sort
of
“noncommutative
plane”].
(ii)
Once
one
has
the
“rigid
standard
integral
structures”
constituted
by
log-
shells
[cf.
Remark
5.10.2,
(iii)],
it
is
natural
to
consider
modifying
these
integral
structures
by
means
of
the
“Gaussian
zeroes”
[i.e.,
the
inverse
of
the
“Gaussian
poles”]
that
appear
in
the
Hodge-Arakelov
theory
of
elliptic
curves
[cf.,
e.g.,
[Mzk6],
§1.1].
From
the
point
of
view
of
this
theory,
this
amounts,
in
effect,
to
considering
the
“crystalline
theta
object”
[cf.
[Mzk7],
§2].
That
is
to
say,
the
mathematical
apparatus
developed
in
the
present
§5
may
be
thought
of
as
a
sort
of
preparatory
step,
relative
to
the
goal
of
constructing
a
“global
MF
∇
-object-
type
version
of
the
crystalline
theta
object”.
This
point
of
view
is
in
line
with
the
point
of
view
of
the
Introduction
to
[Mzk18]
[cf.
also
[Mzk18],
Remark
5.10.2],
together
with
the
fact
that
the
theory
of
the
étale
theta
function
given
in
[Mzk18],
§1,
involves,
in
an
essential
way,
the
theory
of
elliptic
cuspidalizations
[cf.
Remark
2.7.2].
Moreover,
this
point
of
view
is
reminiscent
of
the
discussion
in
[Mzk7],
§2,
of
the
relation
of
crystalline
theta
objects
to
MF
∇
-objects
—
that
is
to
say,
the
crystalline
theta
object
has
many
properties
that
are
similar
to
those
of
an
MF
∇
-
object,
with
the
notable
exception
constituted
by
the
vanishing
of
the
higher
p-
curvatures
despite
the
fact
that
the
Kodaira-Spencer
morphism
is
an
isomorphism
[cf.
[Mzk7],
Remark
2.11].
This
vanishing
of
higher
p-curvatures,
when
viewed
from
the
point
of
view
of
the
theory
of
“VF-patterns”
of
indigenous
bundles
in
[Mzk4],
seems
to
suggest
that,
whereas
the
indigenous
bundles
considered
in
the
p-adic
uniformization
theory
of
[Mzk4]
are
of
“finite
Frobenius
period”
[in
the
sense
that
they
are
fixed,
up
to
isomorphism,
by
some
finite
number
of
applications
of
Frobenius],
the
crystalline
theta
object
may
only
be
equipped
with
an
“MF
∇
-
object
structure”
if
one
allows
for
infinite
Frobenius
periods.
On
the
other
hand,
by
comparison
to
the
Frobenius
morphisms
that
appear
in
the
theory
of
[Mzk4],
the
log-Frobenius
operation
log
certainly
has
the
feel
of
an
operation
of
“infinite
order”.
Moreover,
as
discussed
in
Remark
3.6.5,
the
telecoricity
of
the
mathematical
apparatus
of
Corollary
5.5
may
be
regarded
as
being
analogous
to
nilpotent,
but
non-vanishing
p-curvature.
That
is
to
say:
By
considering
the
crystalline
theta
object
not
in
the
scheme-theoretic
framework
of
[scheme-theoretic!]
Hodge-Arakelov
theory,
but
rather
in
156
SHINICHI
MOCHIZUKI
the
mono-anabelian
framework
of
the
present
paper,
one
obtains
a
theory
in
which
the
“contradiction”
[from
the
point
of
view
of
the
classical
theory
of
MF
∇
-objects]
of
“vanishing
higher
p-curvatures
in
the
presence
of
a
Kodaira-Spencer
isomorphism”
is
naturally
resolved.
The
above
discussion
suggests
that
one
may
refine
the
correspondence
between
“once-punctured
elliptic
curves”
and
“indigenous
bundles”
discussed
in
(i)
as
fol-
lows:
crystalline
theta
objects
in
scheme-theoretic
Hodge-Arakelov
theory
the
theory
of
mono-anabelian
log-Frobenius
compatibility
of
the
present
paper
—
i.e.,
in
essence,
Belyi
cuspidalization
←→
the
scheme-theoretic
aspects
of
indigenous
bundles
[cf.
[Mzk1],
Chapter
I]
←→
the
positive
characteristic
Frobenius-theoretic
aspects
of
indigenous
bundles
—
e.g.,
the
Verschiebung
on
ind.
buns.
[cf.
[Mzk1],
Chapter
II]
Note
that
the
mono-anabelian
theory
of
the
present
paper
depends,
in
an
essential
way,
on
the
technique
of
Belyi
cuspidalization
[cf.
§1].
Since
the
technique
of
elliptic
cuspidalization
[cf.,
e.g.,
the
theory
of
[Mzk18],
§1!]
may
be
thought
of
as
a
sort
of
simplified,
linearized
[cf.
(v)
below]
version
of
the
technique
of
Belyi
cuspidal-
ization,
and
the
Frobenius
action
on
square
differentials
in
the
theory
of
[Mzk1],
Chapter
II,
may
be
identified
with
the
derivative
[i.e.,
a
sort
of
“simplified,
linearized
version”]
of
the
Verschiebung
on
indigenous
bundles,
it
is
natural
to
supplement
the
correspondences
given
above
with
the
following
further
correspondence:
the
theory
of
the
étale
theta
function
given
in
[Mzk18]
—
i.e.,
in
essence,
elliptic
cuspidalization
the
Frobenius
action
on
the
linear
space
of
square
differentials
[cf.
[Mzk1],
Chapter
II]
←→
These
analogies
with
the
theory
of
[Mzk1],
Chapter
II,
suggest
the
following
further
possible
correspondences:
?
hyp.
orbicurves
of
strictly
Belyi
type
←→
nilp.
admissible
ind.
buns.
elliptically
admissible
hyp.
orbicurves
←→
?
nilp.
ordinary
ind.
buns.
[i.e.,
where
all
of
the
hyperbolic
orbicurves
involved
are
defined
over
number
fields
—
cf.
Remark
2.8.3].
At
any
rate,
the
correspondence
with
the
theory
of
Chapters
I,
II
of
[Mzk1]
suggests
strongly
the
existence
of
a
theory
of
canonical
liftings
for
number
fields
equipped
with
a
once-punctured
elliptic
curve
that
is
analogous
to
the
theory
of
Chapter
III
of
[Mzk1].
The
author
hopes
to
develop
such
a
theory
in
a
future
paper.
(iii)
Relative
to
the
discussion
of
“units”
versus
“valuation
monoids”
in
Re-
mark
5.10.2,
(iii),
the
fact
that
the
logarithm
[i.e.,
log-Frobenius]
has
the
effect
of
TOPICS
IN
ABSOLUTE
ANABELIAN
GEOMETRY
III
157
converting
[a
certain
portion
of]
the
“units”
into
a
“new
log-generation
valuation
monoid”
is
very
much
in
line
with
the
“positive
slope”
—
i.e.,
“telecore-theoretic”
—
nature
of
a
uniformizing
MF
∇
-object
[cf.
the
discussion
of
(i),
(ii)].
Indeed,
from
the
point
of
view
of
uniformizations
of
a
Tate
curve
[cf.
the
discussion
of
Remark
2.7.2;
the
discussion
of
the
Introduction
of
[Mzk18]]
the
valuation
monoid
portion
of
an
MLF
corresponds
precisely
to
“slope
zero”,
whereas
the
units
of
an
MLF
correspond
to
“positive
slope”;
a
similar
such
correspondence
also
appears
in
classical
formulations
of
local
class
field
theory.
(iv)
One
important
aspect
of
the
theory
of
the
present
paper
is
that
it
is
only
applicable
to
elliptically
admissible
hyperbolic
orbicurves,
i.e.,
hyperbolic
or-
bicurves
that
are
closely
related
to
a
once-punctured
elliptic
curve.
In
light
of
the
“entrusting
of
local
moduli/ring
structure”
aspect
of
the
theory
of
the
present
paper
discussed
in
(i)
above,
it
seems
reasonable
to
suspect
that
this
special
nature
of
once-punctured
elliptic
curves
[i.e.,
relative
to
the
theory
of
the
present
pa-
per]
may
be
closely
related
to
the
fact
that,
unlike
arbitrary
hyperbolic
orbicurves,
the
moduli
stack
of
once-punctured
elliptic
curves
has
precisely
one
[holomorphic]
dimension
[i.e.,
corresponding
to
the
“one
holomorphic
dimension”
of
a
number
field].
This
“special
nature
of
once-punctured
elliptic
curves”
is
also
reminiscent
of
the
observation
made
in
[Mzk6],
§1.5.2,
to
the
effect
that
it
does
not
appear
possible
[at
least
in
any
immediate
way]
to
generalize
the
scheme-theoretic
Hodge-
Arakelov
theory
of
elliptic
curves
either
to
higher-dimensional
abelian
varieties
or
to
higher
genus
curves.
Moreover,
it
is
reminiscent
of
the
parallelogram-theoretic
reconstruction
algorithms
of
Corollary
2.7,
which,
from
the
point
of
view
of
the
the-
ory
of
[Mzk14],
§2,
may
only
be
performed
canonically
once
one
chooses
some
fixed
“one-dimensional
space
of
square
differentials”
—
a
choice
which
is
not
necessary
in
the
elliptically
admissible
case,
precisely
because
of
the
one-dimensionality
of
the
moduli
of
once-punctured
elliptic
curves.
(v)
Observe
that
the
“arithmetic
Teichmüller
dilations”
discussed
in
Remark
5.10.2,
(iii)
—
which
deform
the
“arithmetic
holomorphic
structure”
—
are
linear
in
nature
[cf.,
e.g.,
the
“unit-linear
Frobenius
functor”].
On
the
other
hand,
the
log-
Frobenius
operation
within
each
“zone
of
arithmetic
holomorphy”
is
“non-linear”,
with
respect
to
both
the
additive
and
multiplicative
structures
of
the
rings
involved.
Indeed,
as
discussed
extensively
in
the
remarks
following
Corollaries
3.6,
3.7
[cf.
also
the
discussion
in
the
latter
half
of
Remark
5.10.2,
(iii)],
the
essential
reason
for
the
introduction
of
mono-anabelian
geometry
in
the
present
paper
is
precisely
the
need
to
deal
with
this
non-linearity.
In
the
classical
theory
of
Teichmüller
deformations
of
Riemann
surfaces,
the
deformations
of
holomorphic
structure
are
linear
[cf.
the
approach
to
this
theory
given
in
[Mzk14],
§2].
On
the
other
hand,
non-linearity
may
be
witnessed
in
classical
Teichmüller
theory
in
the
quadratic
nature
of
the
square
differentials.
Typically,
non-linearity
is
related
to
some
sort
of
“bounded
domain”.
In
the
complex
theory,
the
bounded
nature
of
the
upper
half-plane,
as
well
as
of
Teichmüller
space
itself,
constitute
examples
of
this
phenomenon
—
cf.
the
discussion
of
“Frobenius-invariant
integral
structures”
in
[Mzk4],
Introduction,
§0.4.
In
the
case
of
elliptic
curves,
the
quadratic
nature
of
the
square
differentials
corresponds
precisely
to
the
quadratic
nature
of
the
exponent
that
appears
in
the
classical
series
representation
of
the
theta
function;
moreover,
this
quadratic
cor-
respondence
“Z
n
→
n
2
∈
Z”
is
[unlike
the
linear
correspondence
n
→
c
·
n,
for
c
∈
Z]
bounded
from
below.
Returning
to
the
theory
of
log-shells,
let
us
recall
158
SHINICHI
MOCHIZUKI
that
the
non-linear
log-Frobenius
operation
is
used
precisely
to
achieve
the
cru-
cial
boundedness
[i.e.,
“compactness”]
property
of
log-shells
[cf.
the
discussion
of
Remark
5.10.2!].
Also,
relative
to
the
discussion
of
(ii)
above,
let
us
recall
that
the
goal
of
constructing
a
comparison
isomorphism
between
non-linear
compact
domains
of
function
spaces
formed
one
of
the
key
motivations
for
the
development
of
the
Hodge-Arakelov
theory
of
elliptic
curves
[cf.
[Mzk6],
§1.3.2,
§1.3.3].
(vi)
Relative
to
the
analogy
between
“once-punctured
elliptic
curves
over
num-
ber
fields”
and
“nilpotent
indigenous
bundles”
[cf.
(i)],
it
is
interesting
to
note
that
if
one
thinks
of
the
number
fields
involved
as
“log
number
fields”
—
i.e.,
number
fields
equipped
with
a
finite
set
of
primes
at
which
the
elliptic
curve
is
allowed
to
have
bad
[but
multiplicative!]
reduction
—
then
Siegel’s
classical
finiteness
theorem
[which
implies
the
finiteness
of
the
set
of
isomorphism
classes
of
elliptic
curves
over
a
given
“log
number
field”]
may
be
regarded
as
the
analogue
of
the
finiteness
of
the
Verschiebung
on
indigenous
bundles
given
in
[Mzk1],
Chapter
II,
Theorem
2.3
[which
implies
the
finiteness
of
the
set
of
isomorphism
classes
of
nilpotent
indigenous
bundles
over
a
given
hyperbolic
curve
in
positive
characteristic].
Remark
5.10.4.
The
analogy
with
Frobenius
liftings
that
appears
in
the
discus-
sion
of
Remark
5.10.3
is
interesting
from
the
point
of
view
of
the
theory
of
[Mzk21],
§2
[cf.,
especially
[Mzk21],
Remark
2.9.1].
Indeed,
[Mzk21],
§2,
may
be
thought
of
as
a
theory
concerning
the
issue
of
passing
from
decomposition
groups
to
ring
[i.e.,
additive!]
structures
in
a
p-adic
setting
[cf.
[Mzk21],
Corollary
2.9],
hence
may
be
thought
of
as
a
sort
of
p-adic
analogue
of
the
lemma
of
Uchida
reviewed
in
Proposition
1.3.
TOPICS
IN
ABSOLUTE
ANABELIAN
GEOMETRY
III
159
Appendix:
Complements
on
Complex
Multiplication
In
the
present
Appendix,
we
expose
the
portion
of
the
well-known
theory
of
abelian
varieties
with
complex
multiplication
[cf.,
e.g.,
[Lang-CM],
[Milne-CM],
for
more
details]
that
underlies
the
observation
“(∗
CM
)”
—
i.e.,
roughly
speaking,
to
the
effect
that,
if
p
is
a
prime
number,
then
every
Lubin-Tate
character
on
an
open
subgroup
of
the
inertia
group
of
the
absolute
Galois
group
of
a
p-adic
local
field
arises,
after
possible
restriction
to
an
open
subgroup,
from
a
subquotient
of
the
p-adic
Tate
module
associated
to
an
abelian
variety
with
complex
multiplication
—
related
to
the
author
by
A.
Tamagawa
[cf.
[Mzk20],
Remark
3.8.1].
In
particular,
we
verify
that
this
observation
(∗
CM
)
does
indeed
hold.
[Here,
we
remark
in
passing
that
the
proof
of
(∗
CM
)
given
in
the
present
Appendix
is,
according
to
Tamagawa,
apparently
somewhat
different
from
the
proof
that
he
originally
considered.
Unfor-
tunately,
however,
he
was
unable
to
recall
the
details
of
his
original
argument.]
This
implies
that
the
observation
“(∗
A-qLT
)”
discussed
in
[Mzk20],
Remark
3.8.1,
also
holds,
and
hence,
in
particular,
that
the
hypothesis
of
[Mzk20],
Corollary
3.9,
to
the
effect
that
“either
(∗
A-qLT
)
or
(∗
CM
)
holds”
may
be
eliminated
[i.e.,
that
[Mzk20],
Corollary
3.9,
holds
unconditionally].
On
the
other
hand,
we
conclude
the
present
Appendix
by
observing
that,
in
this
context,
there
still
remains
an
interesting
open
problem
that
could
serve
to
stimulate
further
research.
In
the
following,
we
shall
fix
a
prime
number
p
and
write
Q
for
the
field
of
rational
numbers,
Z
p
for
the
topological
ring
of
p-adic
integers,
Q
p
for
the
topolog-
ical
field
of
p-adic
numbers,
R
for
the
topological
field
of
real
numbers,
C
for
the
topological
field
of
complex
numbers,
ι
:
C
→
C
for
the
automorphism
of
C
given
by
complex
conjugation,
and
Q
alg
⊆
C
for
the
subfield
of
algebraic
numbers.
Also,
we
shall
use
the
notation
“O”
to
denote
the
ring
of
integers
associated
to
a
finite
extension
of
Q
or
Q
p
and
the
notation
“tr
(−)
”
to
denote
the
trace
map
associated
to
a
finite
field
extension
“(−)”.
(CM1)
Fix
a
finite
extension
L
of
degree
d
≥
1
of
Q
p
.
Thus,
L
=
Q
p
(α)
for
some
α
∈
L.
Let
f
(x)
∈
Q
p
[x]
be
a
monic
irreducible
polynomial
such
that
f
(α)
=
0.
If
def
def
d
=
2,
then
set
g(x)
=
x
2
+
1;
if
d
=
2,
then
set
g(x)
=
(x
−
1)(x
−
2)
·
.
.
.
·
(x
−
d).
Thus,
both
f
(x)
∈
Q
p
[x]
and
g(x)
∈
Q[x]
are
of
degree
d.
Then
by
approximating
the
coefficients
of
f
and
g
by
elements
of
Q
at
the
p-adic
and
real
places
of
Q,
we
conclude
that
there
exists
a
monic
polynomial
h(x)
∈
Q[x]
of
degree
d
such
that
the
following
conditions
hold:
(a)
there
exists
an
element
β
∈
L
such
that
h(β)
=
0
and
L
=
Q
p
(β);
(b)
if
d
=
2,
then
the
complex
roots
of
h(x)
are
non-real
and
distinct;
(c)
if
d
=
2,
then
the
complex
roots
of
h(x)
are
real
and
distinct.
Indeed,
(a)
follows
by
arguing
as
in
[Kobl],
pp.
69-70;
(b)
follows
by
considering
the
sign
of
the
discriminant
of
h(x);
(c)
follows
by
considering
the
signs
of
values
of
g(x)
as
x
varies
over
the
real
numbers
in
the
various
intervals
between
roots
of
g(x).
Note
that
it
follows
from
(a)
that
the
polynomial
h(x)
∈
Q[x]
is
irreducible.
Thus,
we
obtain
a
number
field
160
SHINICHI
MOCHIZUKI
def
F
=
Q[x]/(h(x))
such
that
[F
:
Q]
=
d,
and
F
⊗
Q
Q
p
is
isomorphic
to
L.
If
d
=
2,
then
F
is
a
complex
quadratic
extension
of
Q,
hence
admits
an
element
γ
∈
F
\
Q
such
that
γ
2
∈
Q
[which
implies
that
γ
2
<
0,
F
=
Q(γ)].
Next,
let
us
observe
[cf.
[Kobl],
p.
81]
that
1
−
p
3
admits
a
square
root
in
Q
p
.
Thus,
if
d
=
2,
then
the
number
field
F
is
totally
real
and
hence
linearly
disjoint
over
Q
from
the
complex
quadratic
def
extension
K
0
=
Q(λ
0
),
where
λ
20
=
1
−
p
3
.
In
particular,
if
d
=
2,
then
the
number
field
def
K
=
F
·
K
0
is
a
CM
field
[cf.,
e.g.,
[Lang-CM],
Chapter
1,
§2]
of
degree
2d
over
Q.
(CM2)
Suppose
that
d
=
2.
Let
ϕ
0
:
F
→
C
be
an
embedding
such
that
the
imaginary
part
of
ϕ
0
(γ)
is
positive.
Write
ι
F
∈
Gal(F/Q)
for
the
unique
nontrivial
element
of
Gal(F/Q)
[so
ϕ
0
◦
ι
F
=
ι
◦
ϕ
0
].
Then
recall
[cf.,
e.g.,
[Lang-CM],
Chapter
1,
§4]
that
the
complex
torus
C/ϕ
0
(O
F
),
together
with
the
Riemann
form
determined
by
the
pairing
(ξ,
η)
→
tr
F/Q
(ξ
·
ι
F
(η)
·
γ)
∈
Q
[where
ξ,
η
∈
F
],
determine
an
elliptic
curve
E
with
complex
multiplication
by
O
F
,
which
is
defined
over
some
finite
subextension
M
of
ϕ
0
(F
)
in
C.
Now
it
is
immediate
from
the
Main
Theorem
of
Complex
Multiplication
[i.e.,
Shimura
reciprocity
—
cf.,
e.g.,
[Lang-CM],
Chapter
4,
Theorem
1.1;
[Milne-CM],
Theorem
10.1]
that
there
def
exists
an
open
subgroup
H
of
the
inertia
group
⊆
G
M
=
Gal(Q
alg
/M
)
associated
to
some
prime
of
Q
alg
that
divides
p
such
that
H
acts
on
the
p-adic
Tate
module
associated
to
E
via
the
Lubin-Tate
character
associated
to
L.
This
completes
the
proof
of
(∗
CM
)
in
the
case
d
=
2.
(CM3)
Suppose
that
d
=
2.
Let
Φ
0
be
a
collection
of
d
embeddings
K
→
C
of
K
into
the
complex
numbers
such
that
every
embedding
F
→
C
is
obtained
as
the
restriction
of
an
element
of
Φ
0
,
and,
moreover,
the
embeddings
of
Φ
0
map
λ
0
to
a
complex
number
whose
imaginary
part
is
positive.
[Thus,
the
embeddings
of
Φ
0
coincide
on
K
0
.]
Fix
an
element
ϕ
0
∈
Φ
0
.
Thus,
one
verifies
immediately
that
both
Φ
0
and
def
Φ
=
{ϕ
0
}
∪
{ι
◦
ϕ
|
ϕ
0
=
ϕ
∈
Φ
0
}
form
CM
types
of
K
[cf.,
e.g.,
[Lang-CM],
Chapter
1,
§2].
Moreover,
if
we
write
def
Φ
ι
0
=
{ι
◦
ϕ
|
ϕ
∈
Φ
0
},
then
one
verifies
immediately
that
the
set
of
embeddings
K
→
C
[or,
equivalently,
K
→
Q
alg
]
∼
Φ
0
∪
Φ
ι
0
→
Φ
0
×
{id,
ι}
def
[where
id
denotes
the
identity
automorphism
of
C]
admits
a
natural
action
by
G
Q
=
Gal(Q
alg
/Q)
that
preserves
the
product
decomposition
[induced
by
restricting
the
embeddings
in
question
to
F
or
K
0
]
of
the
above
display.
Then
one
verifies
immedi-
def
ately
that
the
subgroup
of
G
Q
that
stabilizes
Φ
0
is
equal
to
G
K
0
=
Gal(Q
alg
/K
0
),
and
hence
that
the
reflex
field
[cf.,
e.g.,
[Lang-CM],
Chapter
1,
§5]
associated
to
(K,
Φ
0
)
is
equal
to
ϕ
0
(K
0
).
On
the
other
hand,
observe
that
our
assumption
that
d
=
2
implies
that
the
cardinalities
[namely,
1
and
d
−
1]
of
the
intersections
Φ
∩
Φ
0
and
Φ
∩
Φ
ι
0
are
distinct.
Thus,
since
the
action
of
any
element
of
G
Q
on
Φ
0
∪
Φ
ι
0
is
TOPICS
IN
ABSOLUTE
ANABELIAN
GEOMETRY
III
161
compatible
with
the
projection
to
the
set
{id,
ι},
one
verifies
immediately
[by
consid-
ering
the
fibers,
i.e.,
Φ
0
and
Φ
ι
0
,
of
this
projection]
that
our
assumption
that
d
=
2
implies
that
an
element
of
G
Q
stabilizes
Φ
if
and
only
if
it
fixes
ϕ
0
.
In
particular,
we
conclude
that
the
reflex
field
[cf.,
e.g.,
[Lang-CM],
Chapter
1,
§5]
associated
to
(K,
Φ)
is
equal
to
ϕ
0
(K).
(CM4)
We
continue
our
analysis
of
the
situation
discussed
in
(CM3).
Write
ι
K
∈
Gal(K/F
)
for
the
unique
nontrivial
element
of
Gal(K/F
)
[so
ϕ
◦
ι
K
=
ι
◦
ϕ,
for
all
ϕ
∈
Φ].
Observe
that
by
approximating
λ
0
relative
to
ϕ
0
and
−λ
0
relative
to
ϕ
∈
Φ
\
{ϕ
0
},
one
may
construct
an
element
λ
∈
K
such
that
the
imaginary
part
of
ϕ(λ)
is
positive
for
all
ϕ
∈
Φ.
Moreover,
by
replacing
λ
by
λ
−
ι
K
(λ),
one
may
assume
without
loss
of
generality
that
ι
K
(λ)
=
−λ.
Next,
recall
[cf.,
e.g.,
[Lang-
CM],
Chapter
1,
§4]
that
the
CM
type
(K,
Φ),
together
with
the
lattice
O
K
⊆
K
and
the
Riemann
form
determined
by
the
pairing
(ξ,
η)
→
tr
K/Q
(ξ
·
ι
K
(η)
·
λ)
∈
Q
[where
ξ,
η
∈
K],
determine
a
polarized
abelian
variety
A
with
complex
multiplication
by
O
K
,
which
is
defined
over
some
finite
subextension
M
of
ϕ
0
(K)
def
in
C.
Next,
write
G
M
=
Gal(Q
alg
/M
),
T
p
(A)
for
the
p-adic
Tate
module
associated
to
A.
Thus,
T
p
(A)
admits
a
natural
structure
of
rank
one
free
O
K
⊗
Z
p
-module,
as
well
as
a
natural
G
M
-action.
In
particular,
since
O
K
⊗
Z
p
∼
=
O
L
⊕
O
L
,
we
thus
conclude
that
T
p
(A)
admits
a
direct
sum
decomposition
T
p
(A)
=
T
⊕
T
as
a
direct
sum
of
rank
one
free
O
L
-modules
T
,
T
.
On
the
other
hand,
let
us
recall
that
the
Main
Theorem
of
Complex
Multiplication
[i.e.,
Shimura
reciprocity
—
cf.,
e.g.,
[Lang-CM],
Chapter
4,
Theorem
1.1;
[Milne-CM],
Theorem
10.1]
allows
one
to
compute
the
Galois
action
of
G
M
on
T
p
(A)
by
means
of
the
reflex
type
norm
applied
to
an
idèle
of
M
.
In
particular,
it
follows
immediately
from
our
construction
of
Φ
from
Φ
0
in
(CM3),
together
with
the
resulting
computation
of
the
associated
reflex
field,
that,
after
possibly
interchanging
T
and
T
,
there
exists
an
open
subgroup
H
of
the
inertia
group
⊆
G
M
associated
to
some
prime
of
Q
alg
that
divides
p
such
that
H
acts
on
T
1
via
the
Lubin-Tate
character
×
χ
LT
:
H
→
O
L
associated
to
L
[i.e.,
in
essence,
via
the
embedding
ϕ
0
]
and
on
T
2
via
the
dual
character
×
χ
∗
LT
:
H
→
O
L
[that
is
to
say,
the
character
determined
by
the
relation
χ
LT
·
χ
∗
LT
=
χ
cycl
,
where
χ
cycl
:
H
→
Z
×
p
is
the
cyclotomic
character,
i.e.,
in
essence,
via
the
product
of
the
embeddings
∈
Φ
\
{ϕ
0
}].
This
completes
the
proof
of
the
observation
(∗
CM
)
[for
arbitrary
d].
(CM5)
The
above
argument
completes
the
proof
of
the
observation
(∗
CM
)
and
hence
also
of
the
observation
(∗
A-qLT
),
of
[Mzk20],
Remark
3.8.1.
On
the
other
hand,
we
conclude
by
observing
that,
in
this
context,
the
following
problem
remains
unresolved:
Let
X
be
a
hyperbolic
curve
over
a
finite
extension
k
of
Q
p
.
Then
is
it
always
the
case
that
the
étale
fundamental
group
of
X
is
of
A-qLT-type
[cf.
[Mzk20],
Definition
3.1,
(v)]?
Here,
we
recall
that,
roughly
speaking,
this
condition
of
being
“of
A-qLT-type”
may
be
described
as
the
condition
that
every
Lubin-Tate
character
on
the
inertia
162
SHINICHI
MOCHIZUKI
subgroup
of
an
open
subgroup
of
the
absolute
Galois
group
of
k
arises,
after
possibly
restricting
to
an
open
subgroup,
from
some
subquotient
of
the
p-adic
Tate
module
of
the
Jacobian
of
a
finite
étale
covering
of
X
[cf.
[Mzk20],
Definition
3.1,
(v),
for
more
details].
Thus,
(∗
A-qLT
)
consists
of
the
assertion
that
this
problem
admits
an
affirmative
answer
whenever
X
admits
a
finite
étale
covering
that,
in
turn,
admits
a
dominant
map
to
a
copy
of
the
projective
line
minus
three
points
over
k.
We
recall
from
[Mzk20],
Remark
3.8.1,
that
(∗
A-qLT
)
is
derived
from
(∗
CM
)
by
using
Belyi
maps.
Thus,
the
above
unresolved
problem
is
particularly
of
interest
in
the
case
of
various
“classes”
of
X
for
which
techniques
involving
Belyi
maps
cannot
be
applied,
e.g.,
the
case
of
proper
X.
Finally,
we
observe
that
this
problem
may
also
be
understood
in
the
context
of
the
general
theme
of
applications
of
Belyi
maps,
i.e.,
in
the
style
of
Belyi
injectivity
or
[André],
Theorems
7.2.1,
7.2.3
[which
may
be
thought
of
as
a
sort
of
p-
adic
version
of
Belyi
injectivity].
In
the
case
of
Belyi
injectivity
or
André’s
results,
a
version
for
arbitrary
hyperbolic
curves
was
obtained,
by
applying
techniques
from
combinatorial
anabelian
ge-
ometry,
in
[HM1],
Theorem
C
[in
the
case
of
Belyi
injectivity]
and
[HM2],
Theorem
B
[in
the
case
of
André’s
results].
On
the
other
hand,
in
the
case
of
the
unresolved
problem
discussed
above,
it
is
not
clear
to
the
author
at
the
present
time
how
to
apply
techniques
from
combinatorial
anabelian
geometry
to
resolve
this
problem.
TOPICS
IN
ABSOLUTE
ANABELIAN
GEOMETRY
III
163
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